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OK, this is linear
algebra lecture nine.
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And this is a key lecture, this
is where we get these ideas
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of linear independence,
when a bunch of vectors are
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independent --
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or dependent,
that's the opposite.
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The space they span.
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A basis for a subspace
or a basis for a vector
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space, that's a central idea.
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And then the dimension
of that subspace.
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So this is the day
that those words
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get assigned clear meanings.
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And emphasize that we talk
about a bunch of vectors
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being independent.
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Wouldn't talk about a
matrix being independent.
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A bunch of vectors
being independent.
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A bunch of vectors
spanning a space.
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A bunch of vectors
being a basis.
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And the dimension
is some number.
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OK, so what are the definitions?
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Can I begin with a fact,
a highly important fact,
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that, I didn't call directly
attention to earlier.
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Suppose I have a matrix and
I look at Ax equals zero.
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Suppose the matrix
has a lot of columns,
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so that n is bigger than m.
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So I'm looking at n equations --
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I mean, sorry, m
equations, a small number
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of equations m,
and more unknowns.
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I have more unknowns
than equations.
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Let me write that down.
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More unknowns than equations.
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More unknown x-s than equations.
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Then the conclusion
is that there's
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something in the null
space of A, other
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than just the zero vector.
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The conclusion is there
are some non-zero x-s
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such that Ax is zero.
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There are some
special solutions.
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And why?
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We know why.
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I mean, it sort of like seems
like a reasonable thing, more
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unknowns than equations,
then it seems reasonable
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that we can solve them.
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But we have a, a clear algorithm
which starts with a system
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and does elimination, gets
the thing into an echelon
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form with some pivots
and pivot columns,
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and possibly some free columns
that don't have pivots.
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And the point is here there
will be some free columns.
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The reason, so the
reason is there must --
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there will be free
variables, at least one.
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That's the reason.
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That we now have this --
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a complete, algorithm, a
complete systematic way to say,
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OK, we take the
system Ax equals zero,
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we row reduce, we identify
the free variables,
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and, since there are n
variables and at most m pivots,
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there will be some free
variables, at least one,
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at least n-m in fact, left over.
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And those variables I can
assign non-zero values to.
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I don't have to
set those to zero.
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I can take them to be
one or whatever I like,
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and then I can solve
for the pivot variables.
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So then it gives me a
solution to Ax equals zero.
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And it's a solution
that isn't all zeros.
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So, that's an important
point that we'll
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use now in this lecture.
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So now I want to say what does
it mean for a bunch of vectors
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to be independent.
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OK.
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So this is like the
background that we know.
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Now I want to speak
about independence.
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OK.
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Let's see.
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I can give you the abstract
definition, and I will,
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but I would also like to
give you the direct meaning.
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So the question is, when
vectors x1, x2 up to --
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Suppose I have n vectors
are independent if.
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Now I have to give you --
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or linearly independent --
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I'll often just say and
write independent for short.
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OK.
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I'll give you the
full definition.
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These are just vectors
in some vector space.
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I can take combinations of them.
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The question is, do any
combinations give zero?
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If some combination
of those vectors
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gives the zero vector,
other than the combination
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of all zeros, then
they're dependent.
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They're independent if no
combination gives the zero
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vector --
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and then I have, I'll have
to put in an except the zero
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combination.
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So what do I mean by that?
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No combination gives
the zero vector.
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Any combination
c1 x1+c2 x2 plus,
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plus cn xn is not zero except
for the zero combination.
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This is when all the c-s,
all the c-s are zero.
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Then of course.
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That combination --
I know I'll get zero.
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But the question is, does any
other combination give zero?
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If not, then the
vectors are independent.
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If some other combination
does give zero,
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the vectors are dependent.
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OK.
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Let's just take examples.
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Suppose I'm in, say, in
two dimensional space.
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OK.
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I give you --
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I'd like to first
take an example --
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let me take an example where
I have a vector and twice that
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vector.
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So that's two vectors, V and 2V.
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Are those dependent
or independent?
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Those are dependent
for sure, right,
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because there's one
vector is twice the other.
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One vector is twice
as long as the other,
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so if the word dependent
means anything,
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these should be dependent.
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And they are.
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And in fact, I would
take two of the first --
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so here's, here is a vector V
and the other guy is a vector
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2V, that's my --
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so there's a vector V1 and
my next vector V2 is 2V1.
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Of course those are
dependent, because two
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of these first vectors minus
the second vector is zero.
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That's a combination of these
two vectors that gives the zero
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vector.
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OK, that was clear.
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Suppose, suppose
I have a vector --
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here's another
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example.
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It's easy example.
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Suppose I have a vector and the
other guy is the zero vector.
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Suppose I have a vector V1
and V2 is the zero vector.
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Then are those vectors
dependent or independent?
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They're dependent again.
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You could say, well, this
guy is zero times that one.
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This one is some
combination of those.
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But let me write
it the other way.
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Let me say -- what combination,
how many V1s and how many V2s
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shall I take to get the zero
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vector?
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If, if V1 is like the vector two
one and V2 is the zero vector,
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zero zero, then I
would like to show
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that some combination of
those gives the zero vector.
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What shall I take?
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How many V1s shall I take?
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Zero of them.
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Yeah, no, take no V1s.
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But how many V2s?
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Six.
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OK.
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Or five.
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Then -- in other words,
the point is if the zero
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vector's in there,
if the zero --
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if one of these vectors
is the zero vector,
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independence is dead, right?
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If one of those vectors is the
zero vector then I could always
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take --
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include that one and
none of the others,
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and I would get the zero answer,
and I would show dependence.
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OK.
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Now, let me, let me
finally draw an example
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where they will be independent.
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Suppose that's V1 and that's V2.
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Those are surely
independent, right?
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Any combination of
V1 and V2, will not
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be zero except, the
zero combination.
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So those would be independent.
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But now let me, let me
stick in a third vector, V3.
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Independent or dependent
now, those three vectors?
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So now n is three here.
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I'm in two dimensional space,
whatever, I'm in the plane.
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I have three vectors that
I didn't draw so carefully.
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I didn't even tell you
what exactly they were.
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But what's this answer on
dependent or independent?
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Dependent.
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How do I know those
are dependent?
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How do I know that some
combination of V1, V2, and V3
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gives me the zero vector?
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I know because of that.
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That's the key
fact that tells me
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that three vectors in the
plane have to be dependent.
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Why's that?
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What's the connection between
the dependence of these three
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vectors and that fact?
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OK.
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So here's the connection.
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I take the matrix A that has
V1 in its first column, V2
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in its second column,
V3 in its third column.
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So it's got three columns.
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And V1 --
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I don't know, that
looks like about two one
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to me.
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V2 looks like it
might be one two.
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V3 looks like it might be maybe
two, maybe two and a half,
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minus one.
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OK.
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Those are my three vectors, and
I put them in the columns of A.
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Now that matrix A
is two by three.
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It fits this pattern, that
where we know we've got extra
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variables, we know we
have some free variables,
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we know that there's
some combination --
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and let me instead of x-s, let
me call them c1, c2, and c3 --
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that gives the zero vector.
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Sorry that my little bit
of art got in the way.
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Do you see the point?
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When I have a matrix,
I'm interested
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in whether its columns are
dependent or independent.
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The columns are
dependent if there
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is something in the null space.
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The columns are
dependent because this,
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this thing in the
null space says
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that c1 of that plus c2 of
that plus c3 of this is zero.
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So in other words, I can go
out some V1, out some more V2,
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back on V3, and end up zero.
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OK.
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So let -- here I've give the
general, abstract definition,
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but let me repeat
that definition --
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this is like repeat --
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let me call them Vs now.
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V1 up to Vn are the
columns of a matrix A.
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In other words,
this is telling me
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that if I'm in m
dimensional space,
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like two dimensional
space in the example,
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I can answer the
dependence-independence
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question directly by
putting those vectors
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in the columns of a matrix.
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They are independent if the
null space of A, of A, is what?
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If I have a bunch of
columns in a matrix,
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I'm looking at
their combinations,
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but that's just A times
the vector of c-s.
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And these columns
will be independent
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if the null space of
A is the zero vector.
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They are dependent if there's
something else in there.
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If there's something else in
the null space, if A times c
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gives the zero vector
for some non-zero vector
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c in the null space.
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Then they're dependent,
because that's
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telling me a combination of the
columns gives the zero column.
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I think you're with
be, because we've seen,
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like, lecture after
lecture, we're
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00:15:40,340 --> 00:15:43,500
looking at the combinations
of the columns and asking,
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do we get zero or don't we?
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And now we're giving
the official name,
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dependent if we do,
independent if we don't.
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So I could express this
in other words now.
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I could say the rank -- what's
the rank in this independent
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case?
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The rank r of the,
of the matrix,
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in the case of
independent columns, is?
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So the columns are independent.
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So how many pivot
columns have I got.
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All n.
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All the columns would
be pivot columns,
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00:16:23,310 --> 00:16:25,980
because free columns
are telling me
256
00:16:25,980 --> 00:16:29,820
that they're a combination
of earlier columns.
257
00:16:29,820 --> 00:16:33,080
So this would be the
case where the rank is n.
258
00:16:33,080 --> 00:16:36,910
This would be the case where
the rank is smaller than n.
259
00:16:39,650 --> 00:16:44,770
So in this case the rank is
n and the null space of A
260
00:16:44,770 --> 00:16:48,750
is only the zero vector.
261
00:16:48,750 --> 00:16:50,800
And no free variables.
262
00:16:50,800 --> 00:16:52,400
No free variables.
263
00:16:56,520 --> 00:16:58,975
And this is the case
yes free variables.
264
00:17:04,300 --> 00:17:09,589
If you'll allow me to stretch
the English language that far.
265
00:17:09,589 --> 00:17:16,290
That's the case where
we have, a combination
266
00:17:16,290 --> 00:17:19,040
that gives the zero column.
267
00:17:19,040 --> 00:17:23,089
I'm often interested in the
case when my vectors are
268
00:17:23,089 --> 00:17:25,560
popped into a matrix.
269
00:17:25,560 --> 00:17:28,349
So the, the definition
over there of independence
270
00:17:28,349 --> 00:17:31,270
didn't talk about any matrix.
271
00:17:31,270 --> 00:17:36,130
The vectors didn't have to be
vectors in N dimensional space.
272
00:17:36,130 --> 00:17:38,530
And I want to give
you some examples
273
00:17:38,530 --> 00:17:41,710
of vectors that
aren't what you think
274
00:17:41,710 --> 00:17:43,970
of immediately as vectors.
275
00:17:43,970 --> 00:17:49,420
But most of the time, this is
-- the vectors we think of are
276
00:17:49,420 --> 00:17:51,200
columns.
277
00:17:51,200 --> 00:17:54,570
And we can put them in a matrix.
278
00:17:54,570 --> 00:17:57,330
And then independence
or dependence
279
00:17:57,330 --> 00:18:01,650
comes back to the null space.
280
00:18:01,650 --> 00:18:02,162
OK.
281
00:18:02,162 --> 00:18:03,620
So that's the idea
of independence.
282
00:18:06,410 --> 00:18:13,770
Can I just, yeah, let
me go on to spanning a
283
00:18:13,770 --> 00:18:19,480
What does it mean for a bunch
of vectors to span a space?
284
00:18:19,480 --> 00:18:20,580
space.
285
00:18:20,580 --> 00:18:23,800
Well, actually, we've
seen it already.
286
00:18:23,800 --> 00:18:28,040
You remember, if we had
a columns in a matrix,
287
00:18:28,040 --> 00:18:33,130
we took all their
combinations and that gave us
288
00:18:33,130 --> 00:18:36,640
the column space.
289
00:18:36,640 --> 00:18:41,250
Those vectors that we started
with span that column space.
290
00:18:41,250 --> 00:18:44,490
So spanning a space means --
291
00:18:44,490 --> 00:18:48,440
so let me move that
important stuff right up.
292
00:18:53,130 --> 00:18:54,050
OK.
293
00:18:54,050 --> 00:19:02,400
So vectors -- let me call
them, say, V1 up to --
294
00:19:02,400 --> 00:19:06,360
call you some different
letter, say Vl --
295
00:19:06,360 --> 00:19:15,680
span a space, a subspace,
or just a vector space
296
00:19:15,680 --> 00:19:24,790
I could say, span a
space means, means
297
00:19:24,790 --> 00:19:40,270
the space consists of all
combinations of those vectors.
298
00:19:46,464 --> 00:19:48,505
That's exactly what we
did with the column space.
299
00:19:51,040 --> 00:19:54,740
So now I could say in shorthand
the columns of a matrix
300
00:19:54,740 --> 00:19:57,090
span the column space.
301
00:19:57,090 --> 00:20:00,750
So you remember it's a bunch of
vectors that have this property
302
00:20:00,750 --> 00:20:05,450
that they span a space, and
actually if I give you a bunch
303
00:20:05,450 --> 00:20:06,480
of vectors and say --
304
00:20:06,480 --> 00:20:10,720
OK, let S be the
space that they span,
305
00:20:10,720 --> 00:20:14,870
in other words let S contain
all their combinations,
306
00:20:14,870 --> 00:20:17,750
that space S will
be the smallest
307
00:20:17,750 --> 00:20:21,830
space with those
vectors in it, right?
308
00:20:21,830 --> 00:20:24,300
Because any space with
those vectors in it
309
00:20:24,300 --> 00:20:28,730
must have all the combinations
of those vectors in it.
310
00:20:28,730 --> 00:20:34,540
And if I stop there, then
I've got the smallest space,
311
00:20:34,540 --> 00:20:37,300
and that's the space
that they span.
312
00:20:37,300 --> 00:20:37,800
OK.
313
00:20:37,800 --> 00:20:39,150
So I'm just --
314
00:20:39,150 --> 00:20:44,150
rather than, needing to say,
take all linear combinations
315
00:20:44,150 --> 00:20:48,950
and put them in a space,
I'm compressing that
316
00:20:48,950 --> 00:20:50,740
into the word span.
317
00:20:53,169 --> 00:20:53,835
Straightforward.
318
00:20:57,520 --> 00:21:00,015
So if I think of a, of
the column space of a
319
00:21:00,015 --> 00:21:00,515
OK. matrix.
320
00:21:03,890 --> 00:21:07,380
I've got their -- so I
start with the columns.
321
00:21:07,380 --> 00:21:08,930
I take all their combinations.
322
00:21:08,930 --> 00:21:10,760
That gives me the columns space.
323
00:21:10,760 --> 00:21:13,440
They span the column space.
324
00:21:13,440 --> 00:21:16,230
Now are they independent?
325
00:21:16,230 --> 00:21:19,620
Maybe yes, maybe no.
326
00:21:19,620 --> 00:21:23,060
It depends on the particular
columns that went into that
327
00:21:23,060 --> 00:21:24,470
matrix.
328
00:21:24,470 --> 00:21:29,700
But obviously I'm highly
interested in a set
329
00:21:29,700 --> 00:21:36,870
of vectors that spans a
space and is independent.
330
00:21:36,870 --> 00:21:41,630
That's, that means like I've
got the right number of vectors.
331
00:21:41,630 --> 00:21:47,580
If I didn't have all of them,
I wouldn't have my whole space.
332
00:21:47,580 --> 00:21:50,290
If I had more than that,
they probably wouldn't --
333
00:21:50,290 --> 00:21:52,450
they wouldn't be independent.
334
00:21:52,450 --> 00:21:56,860
So, like, basis -- and that's
the word that's coming --
335
00:21:56,860 --> 00:21:58,600
is just right.
336
00:21:58,600 --> 00:22:01,310
So here let me put
what that word means.
337
00:22:01,310 --> 00:22:14,270
A basis for a vector space is,
is a, is a sequence of vectors
338
00:22:14,270 --> 00:22:14,770
--
339
00:22:20,050 --> 00:22:27,040
shall I call them V1, V2,
up to let me say Vd now,
340
00:22:27,040 --> 00:22:33,600
I'll stop with that letters
-- that has two properties.
341
00:22:33,600 --> 00:22:36,230
I've got enough vectors
and not too many.
342
00:22:36,230 --> 00:22:39,340
It's a natural idea of a basis.
343
00:22:39,340 --> 00:22:42,570
So a basis is a bunch
of vectors in the space
344
00:22:42,570 --> 00:22:47,200
and it's a so it's a sequence
of vectors with two properties,
345
00:22:47,200 --> 00:22:50,050
with two properties.
346
00:22:54,760 --> 00:22:57,675
One, they are independent.
347
00:23:05,020 --> 00:23:07,990
And two -- you
know what's coming?
348
00:23:07,990 --> 00:23:10,040
-- they span the space.
349
00:23:20,560 --> 00:23:21,880
OK.
350
00:23:21,880 --> 00:23:25,050
Let me take --
351
00:23:25,050 --> 00:23:28,510
so time for examples, of course.
352
00:23:28,510 --> 00:23:32,320
So I'm asking you now
to put definition one,
353
00:23:32,320 --> 00:23:38,030
the definition of independence,
together with definition two,
354
00:23:38,030 --> 00:23:41,800
and let's look at examples,
because this is --
355
00:23:41,800 --> 00:23:44,840
this combination
means the set I've --
356
00:23:44,840 --> 00:23:47,690
of vectors I have is
just right, and the --
357
00:23:47,690 --> 00:23:51,210
so that this idea of a
basis will be central.
358
00:23:51,210 --> 00:23:54,480
I'll always be asking
you now for a basis.
359
00:23:54,480 --> 00:23:58,760
Whenever I look at a
subspace, if I ask you for --
360
00:23:58,760 --> 00:24:00,760
if you give me a basis
for that subspace,
361
00:24:00,760 --> 00:24:02,920
you've told me what it is.
362
00:24:02,920 --> 00:24:07,430
You've told me everything I need
to know about that subspace.
363
00:24:07,430 --> 00:24:10,520
Those -- I take their
combinations and I know that I
364
00:24:10,520 --> 00:24:12,460
need all the combinations.
365
00:24:12,460 --> 00:24:13,210
OK.
366
00:24:13,210 --> 00:24:13,790
Examples.
367
00:24:13,790 --> 00:24:16,640
OK, so examples of a basis.
368
00:24:16,640 --> 00:24:20,060
Let me start with two
dimensional space.
369
00:24:20,060 --> 00:24:22,350
Suppose the space
-- say example.
370
00:24:26,520 --> 00:24:31,150
The space is, oh,
let's make it R^3.
371
00:24:31,150 --> 00:24:35,980
Real three dimensional space.
372
00:24:35,980 --> 00:24:37,330
Give me one basis.
373
00:24:37,330 --> 00:24:38,705
One basis is?
374
00:24:43,970 --> 00:24:49,010
So I want some vectors, because
if I ask you for a basis,
375
00:24:49,010 --> 00:24:53,390
I'm asking you for vectors,
a little list of vectors.
376
00:24:53,390 --> 00:24:57,610
And it should be just right.
377
00:24:57,610 --> 00:25:02,880
So what would be a basis
for three dimensional space?
378
00:25:02,880 --> 00:25:05,540
Well, the first basis that
comes to mind, why don't we
379
00:25:05,540 --> 00:25:07,240
write that down.
380
00:25:07,240 --> 00:25:09,220
The first basis
that comes to mind
381
00:25:09,220 --> 00:25:18,850
is this vector, this
vector, and this vector.
382
00:25:18,850 --> 00:25:19,790
OK.
383
00:25:19,790 --> 00:25:23,540
That's one basis.
384
00:25:23,540 --> 00:25:27,190
Not the only basis, that's
going to be my point.
385
00:25:27,190 --> 00:25:30,930
But let's just see --
yes, that's a basis.
386
00:25:30,930 --> 00:25:33,500
Are, are those
vectors independent?
387
00:25:36,110 --> 00:25:40,120
So that's the like the x, y,
z axes, so if those are not
388
00:25:40,120 --> 00:25:41,780
independent, we're in trouble.
389
00:25:41,780 --> 00:25:43,580
Certainly, they are.
390
00:25:43,580 --> 00:25:48,630
Take a combination c1 of this
vector plus c2 of this vector
391
00:25:48,630 --> 00:25:51,340
plus c3 of that
vector and try to make
392
00:25:51,340 --> 00:25:55,290
it give the zero vector.
393
00:25:55,290 --> 00:25:57,110
What are the c-s?
394
00:25:57,110 --> 00:26:03,080
If c1 of that plus c2 of that
plus c3 of that gives me 0 0 0,
395
00:26:03,080 --> 00:26:05,600
then the c-s are all --
396
00:26:05,600 --> 00:26:06,920
0, right.
397
00:26:06,920 --> 00:26:09,980
So that's the test
for independence.
398
00:26:09,980 --> 00:26:16,210
In the language of matrices,
which was under that board,
399
00:26:16,210 --> 00:26:19,700
I could make those the
columns of a matrix.
400
00:26:19,700 --> 00:26:22,880
Well, it would be
the identity matrix.
401
00:26:22,880 --> 00:26:25,590
Then I would ask, what's the
null space of the identity
402
00:26:25,590 --> 00:26:26,760
matrix?
403
00:26:26,760 --> 00:26:30,560
And you would say it's
only the zero vector.
404
00:26:30,560 --> 00:26:34,740
And I would say, fine, then
the columns are independent.
405
00:26:34,740 --> 00:26:38,370
The only thing -- the identity
times a vector giving zero,
406
00:26:38,370 --> 00:26:40,810
the only vector that
does that is zero.
407
00:26:40,810 --> 00:26:41,980
OK.
408
00:26:41,980 --> 00:26:45,710
Now that's not the only basis.
409
00:26:45,710 --> 00:26:46,660
Far from it.
410
00:26:46,660 --> 00:26:50,680
Tell me another basis, a
second basis, another basis.
411
00:26:57,560 --> 00:27:03,030
So, give me -- well,
I'll just start it out.
412
00:27:03,030 --> 00:27:04,090
One one two.
413
00:27:06,950 --> 00:27:10,780
Two two five.
414
00:27:10,780 --> 00:27:14,420
Suppose I stopped there.
415
00:27:14,420 --> 00:27:20,980
Has that little bunch of
vectors got the properties that
416
00:27:20,980 --> 00:27:24,350
I'm asking for in
a basis for R^3?
417
00:27:24,350 --> 00:27:26,880
We're looking for
a basis for R^3.
418
00:27:26,880 --> 00:27:30,430
Are they independent,
those two column vectors?
419
00:27:30,430 --> 00:27:31,330
Yes.
420
00:27:31,330 --> 00:27:33,610
Do they span R^3?
421
00:27:33,610 --> 00:27:34,340
No.
422
00:27:34,340 --> 00:27:36,440
Our feeling is no.
423
00:27:36,440 --> 00:27:37,550
Our feeling is no.
424
00:27:37,550 --> 00:27:41,640
Our feeling is that there're
some vectors in R3 that
425
00:27:41,640 --> 00:27:44,120
are not combinations of those.
426
00:27:44,120 --> 00:27:44,930
OK.
427
00:27:44,930 --> 00:27:47,830
So suppose I add in --
428
00:27:47,830 --> 00:27:50,390
I need another vector
then, because these two
429
00:27:50,390 --> 00:27:52,021
don't span the space.
430
00:27:52,021 --> 00:27:52,520
OK.
431
00:27:52,520 --> 00:27:56,360
Now it would be foolish for me
to put in three three seven,
432
00:27:56,360 --> 00:27:58,810
right, as the third vector.
433
00:27:58,810 --> 00:27:59,950
That would be a goof.
434
00:27:59,950 --> 00:28:03,380
Because that, if I put
in three three seven,
435
00:28:03,380 --> 00:28:07,680
those vectors would
be dependent, right?
436
00:28:07,680 --> 00:28:09,440
If I put in three
three seven, it
437
00:28:09,440 --> 00:28:12,170
would be the sum
of those two, it
438
00:28:12,170 --> 00:28:15,020
would lie in the
same plane as those.
439
00:28:15,020 --> 00:28:18,590
It wouldn't be independent.
440
00:28:18,590 --> 00:28:21,930
My attempt to create
a basis would be dead.
441
00:28:21,930 --> 00:28:26,420
But if I take -- so
what vector can I take?
442
00:28:26,420 --> 00:28:30,340
I can take any vector
that's not in that plane.
443
00:28:30,340 --> 00:28:31,430
Let me try --
444
00:28:31,430 --> 00:28:33,590
I hope that 3 3 8 would do it.
445
00:28:37,340 --> 00:28:40,470
At least it's not the
sum of those two vectors.
446
00:28:40,470 --> 00:28:44,160
But I believe that's a basis.
447
00:28:44,160 --> 00:28:49,310
And what's the test then,
for that to be a basis?
448
00:28:49,310 --> 00:28:53,160
Because I just picked those
numbers, and if I had picked,
449
00:28:53,160 --> 00:29:02,300
5 7 -14 how would we know do
we have a basis or don't we?
450
00:29:02,300 --> 00:29:05,430
You would put them in
the columns of a matrix,
451
00:29:05,430 --> 00:29:09,280
and you would do
elimination, row reduction --
452
00:29:09,280 --> 00:29:15,340
and you would see do you
get any free variables
453
00:29:15,340 --> 00:29:18,500
or are all the
columns pivot columns.
454
00:29:18,500 --> 00:29:20,590
Well now actually
we have a square --
455
00:29:20,590 --> 00:29:22,760
the matrix would
be three by three.
456
00:29:22,760 --> 00:29:26,260
So, what's the test
on the matrix then?
457
00:29:29,280 --> 00:29:34,790
The matrix -- so in this case,
when my space is R^3 and I have
458
00:29:34,790 --> 00:29:44,740
three vectors, my matrix is
square and what I asking about
459
00:29:44,740 --> 00:29:49,780
that matrix in order for
those columns to be a basis?
460
00:29:49,780 --> 00:29:51,190
So in this --
461
00:29:51,190 --> 00:30:06,380
for R^n, if I have -- n vectors
give a basis if the n by n
462
00:30:06,380 --> 00:30:19,850
matrix with those columns,
with those columns, is what?
463
00:30:19,850 --> 00:30:24,070
What's the requirement
on that matrix?
464
00:30:24,070 --> 00:30:27,010
Invertible, right, right.
465
00:30:27,010 --> 00:30:28,590
The matrix should be invertible.
466
00:30:28,590 --> 00:30:33,187
For a square matrix, that's
the, that's the perfect answer.
467
00:30:33,187 --> 00:30:33,770
Is invertible.
468
00:30:38,260 --> 00:30:43,380
So that's when, that's when the
space is the whole space R^n.
469
00:30:46,140 --> 00:30:50,360
Let me, let me be sure
you're with me here.
470
00:30:50,360 --> 00:30:53,540
Let me remove that.
471
00:30:53,540 --> 00:31:01,580
Are those two vectors a
basis for any space at all?
472
00:31:01,580 --> 00:31:04,020
Is there a vector
space that those really
473
00:31:04,020 --> 00:31:08,540
are a basis for, those, that
pair of vectors, this guy
474
00:31:08,540 --> 00:31:11,580
and this 1, 1 1 2 and 2 2 5?
475
00:31:11,580 --> 00:31:15,360
Is there a space for
which that's a basis?
476
00:31:15,360 --> 00:31:16,220
Sure.
477
00:31:16,220 --> 00:31:21,970
They're independent, so they
satisfy the first requirement,
478
00:31:21,970 --> 00:31:24,740
so what space shall I take
for them to be a basis
479
00:31:24,740 --> 00:31:25,240
of?
480
00:31:25,240 --> 00:31:29,080
What spaces will
they be a basis for?
481
00:31:29,080 --> 00:31:31,180
The one they span.
482
00:31:31,180 --> 00:31:32,350
Their combinations.
483
00:31:32,350 --> 00:31:33,920
It's a plane, right?
484
00:31:33,920 --> 00:31:36,340
It'll be a plane inside R^3.
485
00:31:36,340 --> 00:31:40,930
So if I take this vector
1 1 2, say it goes there,
486
00:31:40,930 --> 00:31:44,240
and this vector 2 2
5, say it goes there,
487
00:31:44,240 --> 00:31:50,797
those are a basis for --
because they span a plane.
488
00:31:50,797 --> 00:31:52,880
And they're a basis for
the plane, because they're
489
00:31:52,880 --> 00:31:53,550
independent.
490
00:31:53,550 --> 00:31:56,800
If I stick in some
third guy, like 3 3 7,
491
00:31:56,800 --> 00:32:01,050
which is in the plane -- suppose
I put in, try to put in 3 3 7,
492
00:32:01,050 --> 00:32:05,740
then the three vectors
would still span the plane,
493
00:32:05,740 --> 00:32:09,180
but they wouldn't be a basis
anymore because they're not
494
00:32:09,180 --> 00:32:12,260
independent anymore.
495
00:32:12,260 --> 00:32:19,420
So, we're looking at
the question of --
496
00:32:19,420 --> 00:32:21,360
again,
497
00:32:21,360 --> 00:32:25,390
OK. the case with
independent columns
498
00:32:25,390 --> 00:32:32,090
is the case where the column
vectors span the column space.
499
00:32:32,090 --> 00:32:35,830
They're independent, so they're
a basis for the column space.
500
00:32:35,830 --> 00:32:36,520
OK.
501
00:32:36,520 --> 00:32:42,490
So now there's one
bit of intuition.
502
00:32:42,490 --> 00:32:46,420
Let me go back to all of R^n.
503
00:32:46,420 --> 00:32:48,150
So I -- where I put 3 3 8.
504
00:32:51,110 --> 00:32:51,740
OK.
505
00:32:51,740 --> 00:32:56,510
The first message is that the
basis is not unique, right.
506
00:32:56,510 --> 00:32:58,090
There's zillions of bases.
507
00:32:58,090 --> 00:33:02,790
I take any invertible
three by three matrix,
508
00:33:02,790 --> 00:33:07,800
its columns are a basis for R^3.
509
00:33:07,800 --> 00:33:11,200
The column space is
R^3, and if those,
510
00:33:11,200 --> 00:33:15,180
if that matrix is invertible,
those columns are independent,
511
00:33:15,180 --> 00:33:17,100
I've got a basis for R^3.
512
00:33:17,100 --> 00:33:19,860
So there're many, many bases.
513
00:33:19,860 --> 00:33:27,730
But there is something in
common for all those bases.
514
00:33:27,730 --> 00:33:32,810
There's something that this
basis shares with that basis
515
00:33:32,810 --> 00:33:35,880
and every other basis for R^3.
516
00:33:35,880 --> 00:33:37,515
And what's that?
517
00:33:40,040 --> 00:33:47,410
Well, you saw it coming, because
when I stopped here and asked
518
00:33:47,410 --> 00:33:51,180
if that was a basis
for R^3, you said no.
519
00:33:51,180 --> 00:33:54,730
And I know that you said
no because you knew there
520
00:33:54,730 --> 00:33:57,410
weren't enough vectors there.
521
00:33:57,410 --> 00:34:03,670
And the great fact is that
there're many, many bases,
522
00:34:03,670 --> 00:34:12,139
but -- let me put in somebody
else, just for variety.
523
00:34:12,139 --> 00:34:14,850
There are many, many
bases, but they all
524
00:34:14,850 --> 00:34:18,639
have the same number of vectors.
525
00:34:18,639 --> 00:34:21,350
If we're talking
about the space R^3,
526
00:34:21,350 --> 00:34:25,040
then that number of
vectors is three.
527
00:34:25,040 --> 00:34:27,650
If we're talking
about the space R^n,
528
00:34:27,650 --> 00:34:31,320
then that number
of vectors is n.
529
00:34:31,320 --> 00:34:36,389
If we're talking about
some other space,
530
00:34:36,389 --> 00:34:41,570
the column space of some matrix,
or the null space of some
531
00:34:41,570 --> 00:34:45,420
matrix, or some other space
that we haven't even thought
532
00:34:45,420 --> 00:34:52,820
of, then that still is
true that every basis --
533
00:34:52,820 --> 00:34:57,270
that there're lots of bases but
every basis has the same number
534
00:34:57,270 --> 00:34:57,950
of vectors.
535
00:34:57,950 --> 00:35:02,000
Let me write that
great fact down.
536
00:35:02,000 --> 00:35:07,940
Every basis --
we're given a space.
537
00:35:07,940 --> 00:35:10,680
Given a space.
538
00:35:13,950 --> 00:35:18,840
R^3 or R^n or some other column
space of a matrix or the null
539
00:35:18,840 --> 00:35:21,790
space of a matrix or
some other vector space.
540
00:35:21,790 --> 00:35:25,200
Then the great fact
is that every basis
541
00:35:25,200 --> 00:35:40,365
for this, for the space has
the same number of vectors.
542
00:35:47,750 --> 00:35:50,770
If one basis has six vectors,
then every other basis
543
00:35:50,770 --> 00:35:52,770
has six vectors.
544
00:35:52,770 --> 00:35:56,050
So that number six
is telling me like
545
00:35:56,050 --> 00:35:59,250
it's telling me how
big is the space.
546
00:35:59,250 --> 00:36:01,640
It's telling me
how many vectors do
547
00:36:01,640 --> 00:36:04,650
I have to have to have a basis.
548
00:36:04,650 --> 00:36:08,440
And of course we're
seeing it this way.
549
00:36:08,440 --> 00:36:10,900
That number six, if
we had seven vectors,
550
00:36:10,900 --> 00:36:13,150
then we've got too many.
551
00:36:13,150 --> 00:36:17,070
If we have five vectors
we haven't got enough.
552
00:36:17,070 --> 00:36:21,860
Sixes are like just right
for whatever space that is.
553
00:36:21,860 --> 00:36:24,600
And what do we call that number?
554
00:36:24,600 --> 00:36:29,550
That number is -- now I'm ready
for the last definition today.
555
00:36:29,550 --> 00:36:33,650
It's the dimension
of that space.
556
00:36:33,650 --> 00:36:37,410
So every basis for a space has
the same number of vectors in
557
00:36:37,410 --> 00:36:37,910
it.
558
00:36:37,910 --> 00:36:41,860
Not the same vectors,
all sorts of bases --
559
00:36:41,860 --> 00:36:44,850
but the same number of
vectors is always the same,
560
00:36:44,850 --> 00:36:47,010
and that number
is the dimension.
561
00:36:47,010 --> 00:36:47,900
This is definitional.
562
00:36:50,780 --> 00:36:58,245
This number is the
dimension of the space.
563
00:37:03,390 --> 00:37:03,890
OK.
564
00:37:06,670 --> 00:37:08,417
OK.
565
00:37:08,417 --> 00:37:09,375
Let's do some examples.
566
00:37:12,930 --> 00:37:14,410
Because now we've
got definitions.
567
00:37:14,410 --> 00:37:17,550
Let me repeat the four
things, the four words that
568
00:37:17,550 --> 00:37:19,270
have now got defined.
569
00:37:19,270 --> 00:37:23,320
Independence, that looks
at combinations not
570
00:37:23,320 --> 00:37:24,560
being zero.
571
00:37:24,560 --> 00:37:27,700
Spanning, that looks at
all the combinations.
572
00:37:27,700 --> 00:37:30,580
Basis, that's the
one that combines
573
00:37:30,580 --> 00:37:32,360
independence and spanning.
574
00:37:32,360 --> 00:37:36,460
And now we've got the idea
of the dimension of a space.
575
00:37:36,460 --> 00:37:40,470
It's the number of
vectors in any basis,
576
00:37:40,470 --> 00:37:43,160
because all bases
have the same number.
577
00:37:43,160 --> 00:37:44,580
OK.
578
00:37:44,580 --> 00:37:47,750
Let's take examples.
579
00:37:47,750 --> 00:37:53,790
Suppose I take, my space
is -- examples now --
580
00:37:53,790 --> 00:37:58,170
space is the, say, the
column space of this matrix.
581
00:37:58,170 --> 00:38:00,240
Let me write down a matrix.
582
00:38:00,240 --> 00:38:06,370
1 1 1, 2 1 2, and I'll
-- just to make it clear,
583
00:38:06,370 --> 00:38:12,430
I'll take the sum there, 3 2 3,
and let me take the sum of all
584
00:38:12,430 --> 00:38:15,050
-- oh, let me put
in one -- yeah,
585
00:38:15,050 --> 00:38:18,530
I'll put in one one one again.
586
00:38:18,530 --> 00:38:20,300
OK.
587
00:38:20,300 --> 00:38:21,260
So that's four vectors.
588
00:38:24,710 --> 00:38:29,310
OK, do they span the column
space of that matrix?
589
00:38:29,310 --> 00:38:35,290
Let me repeat, do they span the
column space of that matrix?
590
00:38:35,290 --> 00:38:37,700
By definition, that's
what the column space --
591
00:38:37,700 --> 00:38:39,280
Yes. where it comes from.
592
00:38:39,280 --> 00:38:41,470
Are they a basis for
the column space?
593
00:38:41,470 --> 00:38:43,920
Are they independent?
594
00:38:43,920 --> 00:38:45,860
No, they're not independent.
595
00:38:45,860 --> 00:38:49,460
There's something
in that null space.
596
00:38:49,460 --> 00:38:55,140
Maybe we can -- so let's look
at the null space of the matrix.
597
00:38:55,140 --> 00:38:58,490
Tell me a vector that's in
the null space of that matrix.
598
00:39:01,090 --> 00:39:05,320
So I'm looking for some vector
that combines those columns
599
00:39:05,320 --> 00:39:08,350
and produces the zero column.
600
00:39:08,350 --> 00:39:10,680
Or in other words, I'm
looking for solutions
601
00:39:10,680 --> 00:39:12,280
to A X equals zero.
602
00:39:12,280 --> 00:39:16,930
So tell me a vector
in the null space.
603
00:39:16,930 --> 00:39:21,070
Maybe -- well, this was, this
column was that one plus that
604
00:39:21,070 --> 00:39:25,030
one, so maybe if I have one of
those and minus one of those
605
00:39:25,030 --> 00:39:26,560
that would be a
vector in the null
606
00:39:26,560 --> 00:39:27,060
space.
607
00:39:29,450 --> 00:39:33,440
So, you've already told me now,
are those vectors independent,
608
00:39:33,440 --> 00:39:38,140
the answer is -- those column
vectors, the answer is --
609
00:39:38,140 --> 00:39:38,670
no.
610
00:39:38,670 --> 00:39:39,170
Right?
611
00:39:39,170 --> 00:39:40,770
They're not independent.
612
00:39:40,770 --> 00:39:44,060
Because -- you knew they
weren't independent.
613
00:39:44,060 --> 00:39:47,060
Anyway, minus one
of this minus one
614
00:39:47,060 --> 00:39:50,530
of this plus one of this zero
of that is the zero vector.
615
00:39:54,120 --> 00:39:55,660
OK, so they're not independent.
616
00:39:55,660 --> 00:39:55,720
OK.
617
00:39:55,720 --> 00:39:57,500
They span, but they're
not independent.
618
00:39:57,500 --> 00:40:04,560
Tell me a basis for
that column space.
619
00:40:04,560 --> 00:40:06,300
What's a basis for
the column space?
620
00:40:06,300 --> 00:40:09,220
These are all the questions that
the homework asks, the quizzes
621
00:40:09,220 --> 00:40:11,800
ask, the final exam will ask.
622
00:40:11,800 --> 00:40:17,710
Find a basis for the column
space of this matrix.
623
00:40:17,710 --> 00:40:20,280
OK.
624
00:40:20,280 --> 00:40:22,500
Now there's many
answers, but give me
625
00:40:22,500 --> 00:40:25,590
the most natural answer.
626
00:40:25,590 --> 00:40:29,670
Columns one and two.
627
00:40:29,670 --> 00:40:31,170
Columns one and two.
628
00:40:31,170 --> 00:40:32,410
That's the natural answer.
629
00:40:32,410 --> 00:40:35,320
Those are the pivot
columns, because, I mean,
630
00:40:35,320 --> 00:40:37,140
we s- we begin systematically.
631
00:40:37,140 --> 00:40:39,300
We look at the first
column, it's OK.
632
00:40:39,300 --> 00:40:41,330
We can put that in the basis.
633
00:40:41,330 --> 00:40:43,550
We look at the second
column, it's OK.
634
00:40:43,550 --> 00:40:46,300
We can put that in the basis.
635
00:40:46,300 --> 00:40:48,960
The third column we
can't put in the basis.
636
00:40:48,960 --> 00:40:54,070
The fourth column
we can't, again.
637
00:40:54,070 --> 00:40:57,570
So the rank of the matrix is --
638
00:40:57,570 --> 00:40:59,850
what's the rank of our matrix?
639
00:40:59,850 --> 00:41:01,210
Two.
640
00:41:01,210 --> 00:41:02,000
Two.
641
00:41:02,000 --> 00:41:06,610
And, and now that rank is also
-- we also have another word.
642
00:41:06,610 --> 00:41:08,710
We, we have a
great theorem here.
643
00:41:08,710 --> 00:41:23,060
The rank of A, that rank r,
is the number of pivot columns
644
00:41:23,060 --> 00:41:25,340
and it's also --
645
00:41:25,340 --> 00:41:27,890
well, so now please
use my new word.
646
00:41:32,640 --> 00:41:34,700
This, it's the number
two, of course,
647
00:41:34,700 --> 00:41:41,790
two is the rank
of my matrix, it's
648
00:41:41,790 --> 00:41:44,440
the number of pivot columns,
those pivot columns form
649
00:41:44,440 --> 00:41:49,104
a basis, of course,
so what's two?
650
00:41:49,104 --> 00:41:49,895
It's the dimension.
651
00:41:52,820 --> 00:41:55,580
The rank of A, the
number of pivot columns,
652
00:41:55,580 --> 00:42:02,555
is the dimension of
the column space.
653
00:42:07,450 --> 00:42:08,840
Of course, you say.
654
00:42:08,840 --> 00:42:10,670
It had to be.
655
00:42:10,670 --> 00:42:12,000
Right.
656
00:42:12,000 --> 00:42:17,440
But just watch, look
for one moment at the,
657
00:42:17,440 --> 00:42:20,240
the language, the
way the English words
658
00:42:20,240 --> 00:42:21,810
get involved here.
659
00:42:21,810 --> 00:42:28,610
I take the rank of a matrix,
the rank of a matrix.
660
00:42:28,610 --> 00:42:34,580
It's a number of columns
and it's the dimension of --
661
00:42:34,580 --> 00:42:37,810
not the dimension of the matrix,
that's what I want to say.
662
00:42:37,810 --> 00:42:43,890
It's the dimension of a space,
a subspace, the column space.
663
00:42:43,890 --> 00:42:46,800
Do you see, I don't
take the dimension of A.
664
00:42:46,800 --> 00:42:49,600
That's not what I want.
665
00:42:49,600 --> 00:42:53,190
I'm looking for the dimension
of the column space of A.
666
00:42:53,190 --> 00:42:56,050
If you use those words right,
it shows you've got the idea
667
00:42:56,050 --> 00:42:57,400
right.
668
00:42:57,400 --> 00:42:59,170
Similarly here.
669
00:42:59,170 --> 00:43:03,430
I don't talk about the
rank of a subspace.
670
00:43:03,430 --> 00:43:05,440
It's a matrix that has a rank.
671
00:43:05,440 --> 00:43:08,040
I talk about the
rank of a matrix.
672
00:43:08,040 --> 00:43:11,750
And the beauty is that
these definitions just
673
00:43:11,750 --> 00:43:14,160
merge so that the
rank of a matrix
674
00:43:14,160 --> 00:43:16,170
is the dimension of
its column space.
675
00:43:16,170 --> 00:43:18,840
And in this example it's two.
676
00:43:18,840 --> 00:43:22,570
And then the further
question is, what's a basis?
677
00:43:22,570 --> 00:43:25,450
And the first two
columns are a basis.
678
00:43:25,450 --> 00:43:27,500
Tell me another basis.
679
00:43:27,500 --> 00:43:29,710
Another basis for
the columns space.
680
00:43:29,710 --> 00:43:31,390
You see I just keep
hammering away.
681
00:43:31,390 --> 00:43:35,140
I apologize, but it's,
I have to be sure you
682
00:43:35,140 --> 00:43:36,660
have the idea of basis.
683
00:43:36,660 --> 00:43:40,940
Tell me another basis
for the column space.
684
00:43:40,940 --> 00:43:47,410
Well, you could take
columns one and three.
685
00:43:47,410 --> 00:43:49,930
That would be a basis
for the column space.
686
00:43:49,930 --> 00:43:53,220
Or columns two and
three would be a basis.
687
00:43:53,220 --> 00:43:55,660
Or columns two and four.
688
00:43:55,660 --> 00:43:58,871
Or tell me another basis that's
not made out of those columns
689
00:43:58,871 --> 00:43:59,370
at all?
690
00:44:03,290 --> 00:44:07,180
So -- I guess I'm giving you
infinitely many possibilities,
691
00:44:07,180 --> 00:44:11,150
so I can't expect a
unanimous answer here.
692
00:44:11,150 --> 00:44:14,770
I'll tell you -- but let's
look at another basis, though.
693
00:44:14,770 --> 00:44:17,836
I'll just -- because it's
only one out of zillions,
694
00:44:17,836 --> 00:44:19,960
I'm going to put it down
and I'm going to erase it.
695
00:44:19,960 --> 00:44:26,680
Another basis for the
column space would be --
696
00:44:26,680 --> 00:44:27,879
let's see.
697
00:44:27,879 --> 00:44:29,670
I'll put in some things
that are not there.
698
00:44:29,670 --> 00:44:33,830
Say, oh well, just to make
it -- my life easy, 2 2 2.
699
00:44:33,830 --> 00:44:37,040
That's in the column space.
700
00:44:37,040 --> 00:44:40,380
And, that was sort of obvious.
701
00:44:40,380 --> 00:44:43,458
Let me take the sum
of those, say 6 4 6.
702
00:44:46,810 --> 00:44:52,040
Or the sum of all of the
columns, 7 5 7, why not.
703
00:44:52,040 --> 00:44:54,910
That's in the column space.
704
00:44:54,910 --> 00:44:59,540
Those are independent and
I've got the number right,
705
00:44:59,540 --> 00:45:01,290
I've got two.
706
00:45:01,290 --> 00:45:02,940
Actually, this is a key point.
707
00:45:05,890 --> 00:45:09,170
If you know the dimension of
the space you're working with,
708
00:45:09,170 --> 00:45:14,480
and we know that this column
-- we know that the dimension,
709
00:45:14,480 --> 00:45:19,430
DIM, the dimension of
the column space is two.
710
00:45:23,730 --> 00:45:30,120
If you know the
dimension, then --
711
00:45:30,120 --> 00:45:33,670
and we have a couple of
vectors that are independent,
712
00:45:33,670 --> 00:45:35,710
they'll automatically be a
713
00:45:35,710 --> 00:45:36,550
basis.
714
00:45:36,550 --> 00:45:38,890
If we've got the number
of vectors right,
715
00:45:38,890 --> 00:45:43,760
two vectors in this case,
then if they're independent,
716
00:45:43,760 --> 00:45:47,037
they can't help
but span the space.
717
00:45:47,037 --> 00:45:48,620
Because if they
didn't span the space,
718
00:45:48,620 --> 00:45:52,130
there'd be a third guy
to help span the space,
719
00:45:52,130 --> 00:45:54,160
but it couldn't be independent.
720
00:45:54,160 --> 00:45:58,190
So, it just has
to be independent
721
00:45:58,190 --> 00:46:01,640
if we've got the numbers right.
722
00:46:01,640 --> 00:46:02,590
And they span.
723
00:46:02,590 --> 00:46:03,280
OK.
724
00:46:03,280 --> 00:46:04,100
Very good.
725
00:46:04,100 --> 00:46:06,060
So you got the
dimension of a space.
726
00:46:06,060 --> 00:46:08,870
So this was another basis
that I just invented.
727
00:46:08,870 --> 00:46:09,470
OK.
728
00:46:09,470 --> 00:46:16,270
Now, now I get to ask
about the null space.
729
00:46:16,270 --> 00:46:18,320
What's the dimension
of the null space?
730
00:46:18,320 --> 00:46:20,700
So we, we got a
great fact there,
731
00:46:20,700 --> 00:46:28,560
the dimension of the
column space is the rank.
732
00:46:28,560 --> 00:46:30,760
Now I want to ask you
about the null space.
733
00:46:30,760 --> 00:46:34,190
That's the other
part of the lecture,
734
00:46:34,190 --> 00:46:38,320
and it'll go on to
the next lecture.
735
00:46:38,320 --> 00:46:40,320
OK.
736
00:46:40,320 --> 00:46:44,230
So we know the dimension of the
column space is two, the rank.
737
00:46:44,230 --> 00:46:46,280
What about the null space?
738
00:46:46,280 --> 00:46:47,870
This is a vector
in the null space.
739
00:46:47,870 --> 00:46:49,770
Are there other vectors
in the null space?
740
00:46:52,280 --> 00:46:53,530
Yes or no?
741
00:46:53,530 --> 00:46:54,870
Yes.
742
00:46:54,870 --> 00:46:58,820
So this isn't a basis because
it's doesn't span, right?
743
00:46:58,820 --> 00:47:01,970
There's more in the null
space than we've got so far.
744
00:47:01,970 --> 00:47:04,570
I need another vector at least.
745
00:47:04,570 --> 00:47:08,790
So tell me another
vector in the null space.
746
00:47:08,790 --> 00:47:12,290
Well, the natural choice, the
choice you naturally think of
747
00:47:12,290 --> 00:47:16,140
is I'm going on to
the fourth column,
748
00:47:16,140 --> 00:47:20,210
I'm letting that free
variable be a one,
749
00:47:20,210 --> 00:47:23,750
and that free variable
be a zero, and I'm asking
750
00:47:23,750 --> 00:47:26,130
is that fourth
column a combination
751
00:47:26,130 --> 00:47:27,410
of my pivot columns?
752
00:47:27,410 --> 00:47:28,670
Yes, it is.
753
00:47:28,670 --> 00:47:31,650
And it's -- that will do.
754
00:47:35,730 --> 00:47:37,820
So what I've written
there are actually the two
755
00:47:37,820 --> 00:47:39,860
special solutions, right?
756
00:47:39,860 --> 00:47:44,520
I took the two free
variables, free and free.
757
00:47:44,520 --> 00:47:50,410
I gave them the values 1 0
or 0 I figured out the rest.
758
00:47:50,410 --> 00:47:53,640
So do you see, let me
just say it in words.
759
00:47:53,640 --> 00:47:58,000
This vector, these vectors in
the null space are telling me,
760
00:47:58,000 --> 00:48:00,190
they're telling me
the combinations
761
00:48:00,190 --> 00:48:02,610
of the columns that give zero.
762
00:48:02,610 --> 00:48:08,770
They're telling me in what way
the, the columns are dependent.
763
00:48:08,770 --> 00:48:10,770
That's what the
null space is doing.
764
00:48:10,770 --> 00:48:13,890
Have I got enough now?
765
00:48:13,890 --> 00:48:15,780
And what's the null space now?
766
00:48:15,780 --> 00:48:18,540
We have to think
about the null space.
767
00:48:18,540 --> 00:48:20,740
These are two vectors
in the null space.
768
00:48:20,740 --> 00:48:21,790
They're independent.
769
00:48:21,790 --> 00:48:24,620
Are they a basis
for the null space?
770
00:48:24,620 --> 00:48:27,650
What's the dimension
of the null space?
771
00:48:27,650 --> 00:48:30,100
You see that those questions
just keep coming up all the
772
00:48:30,100 --> 00:48:31,120
time.
773
00:48:31,120 --> 00:48:34,500
Are they a basis
for the null space?
774
00:48:34,500 --> 00:48:36,940
You can tell me the answer
even though we haven't
775
00:48:36,940 --> 00:48:39,460
written out a proof of that.
776
00:48:39,460 --> 00:48:40,190
Can you?
777
00:48:40,190 --> 00:48:41,050
Yes or no?
778
00:48:41,050 --> 00:48:44,510
Do these two special
solutions form
779
00:48:44,510 --> 00:48:46,400
a basis for the null space?
780
00:48:46,400 --> 00:48:48,790
In other words,
does the null space
781
00:48:48,790 --> 00:48:52,390
consist of all combinations
of those two guys?
782
00:48:52,390 --> 00:48:54,110
Yes or no?
783
00:48:54,110 --> 00:48:55,180
Yes.
784
00:48:55,180 --> 00:48:56,880
Yes.
785
00:48:56,880 --> 00:48:58,970
The null space is
two dimensional.
786
00:48:58,970 --> 00:49:01,500
The null space, the
dimension of the null space,
787
00:49:01,500 --> 00:49:03,530
is the number of free variables.
788
00:49:03,530 --> 00:49:09,330
So the dimension
of the null space
789
00:49:09,330 --> 00:49:12,065
is the number of free variables.
790
00:49:17,260 --> 00:49:21,420
And at the last second,
give me the formula.
791
00:49:21,420 --> 00:49:24,490
This is then the key
formula that we know.
792
00:49:24,490 --> 00:49:29,740
How many free variables are
there in terms of R, the rank,
793
00:49:29,740 --> 00:49:35,020
m -- the number of rows,
n, the number of columns?
794
00:49:35,020 --> 00:49:36,740
What do we get?
795
00:49:36,740 --> 00:49:42,360
We have n columns, r of
them are pivot columns,
796
00:49:42,360 --> 00:49:48,200
so n-r is the number of free
columns, free variables.
797
00:49:48,200 --> 00:49:51,950
And now it's the dimension
of the null space.
798
00:49:51,950 --> 00:49:53,000
OK.
799
00:49:53,000 --> 00:49:53,950
That's great.
800
00:49:53,950 --> 00:49:57,280
That's the key spaces, their
bases, and their dimensions.
801
00:49:57,280 --> 00:49:58,830
Thanks.