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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
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ideas of linear independence,
when a bunch of vectors are
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independent -- 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 space,
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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 get assigned clear
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meanings.
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And emphasize that we talk
about a bunch of vectors being
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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, so that n is bigger
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than m.
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So I'm looking at n equations
-- I mean, sorry,
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m equations,
a small number of equations m,
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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 something in the null
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space of A, other than just the
zero vector.
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The conclusion is there are
some non-zero x-s such that Ax
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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,
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more unknowns than equations,
then it seems reasonable that
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we can solve them.
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But we have a,
a clear algorithm which starts
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with a system and does
elimination, gets the thing into
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an echelon 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 -- there will be free
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variables, at least one.
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That's the reason.
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That we now have this --
a complete, algorithm,
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a complete systematic way to
say, OK, we take the system Ax
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equals zero, 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 use now in this
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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,
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x2 up to --
Suppose I have n vectors are
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independent if.
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Now I have to give you -- or
linearly independent -- I'll
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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 gives the zero vector,
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other than the combination of
all zeros, then they're
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dependent.
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They're independent if no
combination gives the zero
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vector -- and then I have,
I'll have to put in an except
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the zero 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, plus cn xn is not zero
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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
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zero?
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If not, then the vectors are
independent.
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If some other combination does
give zero, the vectors are
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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 -- I'd like to first
take an example -- let me take
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an example where I have a vector
and twice that 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, because there's one
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vector is twice the other.
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One vector is twice as long as
the other, so if the word
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dependent means anything,
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
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other guy is a vector 2V,
that's my -- so there's a
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vector V1 and my next vector V2
is 2V1.
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Of course those are dependent,
because two of these first
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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 -- 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
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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,
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how many V1s and how many V2s
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
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vector, zero zero,
then I would like to show that
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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, independence is
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dead, right?
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If one of those vectors is the
zero vector then I could always
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take -- include that one and
none of the others,
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and I would get the zero
answer, and I would show
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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 be zero except,
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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,
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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,
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V2, and V3 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 that three vectors in the
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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,
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V2 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 -- 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
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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
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extra variables,
we know we have some free
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variables, we know that there's
some combination --
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and let me instead of x-s,
let me call them c1,
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c2, and c3 -- 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 in whether its
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columns are dependent or
independent.
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The columns are dependent if
there is something in the null
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space.
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The columns are dependent
because this,
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this thing in the null space
says that c1 of that plus c2 of
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that plus c3 of this is zero.
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So in other words,
I can go out some V1,
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out some more V2,
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 -- this is like
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repeat -- 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 that if I'm
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in m dimensional space,
like two dimensional space in
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the example, I can answer the
dependence-independence question
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directly by putting those
vectors in the columns of a
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matrix.
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They are independent if the
null space of A,
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of A, is what?
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If I have a bunch of columns in
a matrix, I'm looking at their
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combinations,
but that's just A times the
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vector of c-s.
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And these columns will be
independent if the null space of
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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,
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if A times c 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 telling me a
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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 looking at the
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combinations of the columns and
asking, do we get zero or don't
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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, because free columns
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are telling me that they're a
combination of earlier columns.
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So this would be the case where
the rank is n.
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This would be the case where
the rank is smaller than n.
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So in this case the rank is n
and the null space of A is only
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the zero vector.
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And no free variables.
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No free variables.
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And this is the case yes free
variables.
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If you'll allow me to stretch
the English language that far.
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That's the case where we have,
a combination that gives the
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zero column.
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I'm often interested in the
case when my vectors are popped
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into a matrix.
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So the, the definition over
there of independence didn't
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talk about any matrix.
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The vectors didn't have to be
vectors in N dimensional space.
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And I want to give you some
examples of vectors that aren't
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what you think of immediately as
vectors.
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But most of the time,
this is -- the vectors we think
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of are columns.
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And we can put them in a
matrix.
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And then independence or
dependence comes back to the
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null space.
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OK.
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So that's the idea of
independence.
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Can I just, yeah,
let me go on to spanning a
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space.
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What does it mean for a bunch
of vectors to span a space?
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Well, actually,
we've seen it already.
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You remember,
if we had a columns in a
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matrix, we took all their
combinations and that gave us
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the column space.
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Those vectors that we started
with span that column space.
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So spanning a space means -- so
let me move that important stuff
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right up.
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OK.
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00:18:52 --> 00:19:02
So vectors -- let me call them,
say, V1 up to -- call you some
286
00:19:02 --> 00:19:10
different letter,
say Vl -- span a space,
287
00:19:10 --> 00:19:18
a subspace, or just a vector
space I could say,
288
00:19:18 --> 00:19:26
span a space means,
means the space consists of all
289
00:19:26 --> 00:19:34
combinations of those vectors.
290
00:19:34 --> 00:19:39
That's exactly what we did with
the column space.
291
00:19:39 --> 00:19:46
So now I could say in shorthand
the columns of a matrix span the
292
00:19:46 --> 00:19:48
column space.
293
00:19:48 --> 00:19:55
So you remember it's a bunch of
vectors that have this property
294
00:19:55 --> 00:20:01
that they span a space,
and actually if I give you a
295
00:20:01 --> 00:20:07.72
bunch of vectors and say --
OK, let S be the space that
296
00:20:07.72 --> 00:20:13
they span, in other words let S
contain all their combinations,
297
00:20:13 --> 00:20:17
that space S will be the
smallest space with those
298
00:20:17 --> 00:20:18
vectors in it,
right?
299
00:20:18 --> 00:20:24
Because any space with those
vectors in it must have all the
300
00:20:24 --> 00:20:27
combinations of those vectors in
it.
301
00:20:27 --> 00:20:32
And if I stop there,
then I've got the smallest
302
00:20:32 --> 00:20:37
space, and that's the space that
they span.
303
00:20:37 --> 00:20:37
OK.
304
00:20:37 --> 00:20:41
So I'm just -- rather than,
needing to say,
305
00:20:41 --> 00:20:47.25
take all linear combinations
and put them in a space,
306
00:20:47.25 --> 00:20:51
I'm compressing that into the
word span.
307
00:20:51 --> 00:20:54.24
Straightforward.
308
00:20:54.24 --> 00:20:54
OK.
309
00:20:54 --> 00:20:58.78
So if I think of a,
of the column space of a
310
00:20:58.78 --> 00:20:59
matrix.
311
00:20:59 --> 00:21:03
I've got their -- so I start
with the columns.
312
00:21:03 --> 00:21:06
I take all their combinations.
313
00:21:06 --> 00:21:09
That gives me the columns
space.
314
00:21:09 --> 00:21:12
They span the column space.
315
00:21:12 --> 00:21:15
Now are they independent?
316
00:21:15 --> 00:21:18
Maybe yes, maybe no.
317
00:21:18 --> 00:21:23
It depends on the particular
columns that went into that
318
00:21:23 --> 00:21:23
matrix.
319
00:21:23 --> 00:21:28
But obviously I'm highly
interested in a set of vectors
320
00:21:28 --> 00:21:31
that spans a space and is
independent.
321
00:21:31 --> 00:21:37.29
That's, that means like I've
got the right number of vectors.
322
00:21:37.29 --> 00:21:42
If I didn't have all of them,
I wouldn't have my whole space.
323
00:21:42 --> 00:21:48
If I had more than that,
they probably wouldn't --
324
00:21:48 --> 00:21:52
they wouldn't be independent.
325
00:21:52 --> 00:22:00
So, like, basis -- and that's
the word that's coming -- is
326
00:22:00 --> 00:22:02
just right.
327
00:22:02 --> 00:22:08
So here let me put what that
word means.
328
00:22:08 --> 00:22:16
A basis for a vector space is,
is a, is a sequence of vectors
329
00:22:16 --> 00:22:20
--
shall I call them V1,
330
00:22:20 --> 00:22:26
V2, up to let me say Vd now,
I'll stop with that letters --
331
00:22:26 --> 00:22:28
that has two properties.
332
00:22:28 --> 00:22:32
I've got enough vectors and not
too many.
333
00:22:32 --> 00:22:36
It's a natural idea of a basis.
334
00:22:36 --> 00:22:41
So a basis is a bunch of
vectors in the space and it's a
335
00:22:41 --> 00:22:46
so it's a sequence of vectors
with two properties,
336
00:22:46 --> 00:22:49
with two properties.
337
00:22:49 --> 00:22:55
One, they are independent.
338
00:22:55 --> 00:23:02
And two -- you know what's
coming?
339
00:23:02 --> 00:23:07
-- they span the space.
340
00:23:07 --> 00:23:07
OK.
341
00:23:07 --> 00:23:17
Let me take -- so time for
examples, of course.
342
00:23:17 --> 00:23:26
So I'm asking you now to put
definition one,
343
00:23:26 --> 00:23:37
the definition of independence,
together with definition two,
344
00:23:37 --> 00:23:41
and let's look at examples,
because this is -- this
345
00:23:41 --> 00:23:45
combination means the set I've
-- of vectors I have is just
346
00:23:45 --> 00:23:50
right, and the -- so that this
idea of a basis will be central.
347
00:23:50 --> 00:23:54
I'll always be asking you now
for a basis.
348
00:23:54 --> 00:23:58
Whenever I look at a subspace,
if I ask you for -- if you give
349
00:23:58 --> 00:24:02
me a basis for that subspace,
you've told me what it is.
350
00:24:02 --> 00:24:05
You've told me everything I
need to know about that
351
00:24:05 --> 00:24:06
subspace.
352
00:24:06 --> 00:24:09
Those -- I take their
combinations and I know that I
353
00:24:09 --> 00:24:11
need all the combinations.
354
00:24:11 --> 00:24:12
OK.
355
00:24:12 --> 00:24:13
Examples.
356
00:24:13 --> 00:24:17
OK, so examples of a basis.
357
00:24:17 --> 00:24:23.02
Let me start with two
dimensional space.
358
00:24:23.02 --> 00:24:27
Suppose the space -- say
example.
359
00:24:27 --> 00:24:32.44
The space is,
oh, let's make it R^3.
360
00:24:32.44 --> 00:24:36
Real three dimensional space.
361
00:24:36 --> 00:24:39
Give me one basis.
362
00:24:39 --> 00:24:42
One basis is?
363
00:24:42 --> 00:24:47
So I want some vectors,
because if I ask you for a
364
00:24:47 --> 00:24:52
basis, I'm asking you for
vectors, a little list of
365
00:24:52 --> 00:24:53
vectors.
366
00:24:53 --> 00:24:56
And it should be just right.
367
00:24:56 --> 00:25:01
So what would be a basis for
three dimensional space?
368
00:25:01 --> 00:25:05
Well, the first basis that
comes to mind,
369
00:25:05 --> 00:25:10
why don't we write that down.
370
00:25:10 --> 00:25:15
The first basis that comes to
mind is this vector,
371
00:25:15 --> 00:25:18
this vector,
and this vector.
372
00:25:18 --> 00:25:18
OK.
373
00:25:18 --> 00:25:20
That's one basis.
374
00:25:20 --> 00:25:25
Not the only basis,
that's going to be my point.
375
00:25:25 --> 00:25:29
But let's just see -- yes,
that's a basis.
376
00:25:29 --> 00:25:34
Are, are those vectors
independent?
377
00:25:34 --> 00:25:38
So that's the like the x,
y, z axes, so if those are not
378
00:25:38 --> 00:25:40
independent, we're in trouble.
379
00:25:40 --> 00:25:42
Certainly, they are.
380
00:25:42 --> 00:25:46
Take a combination c1 of this
vector plus c2 of this vector
381
00:25:46 --> 00:25:50
plus c3 of that vector and try
to make it give the zero vector.
382
00:25:50 --> 00:25:52
What are the c-s?
383
00:25:52 --> 00:25:56
If c1 of that plus c2 of that
plus c3 of that gives me 0 0 0,
384
00:25:56 --> 00:25:59
then the c-s are all -- 0,
right.
385
00:25:59 --> 00:26:03
So that's the test for
independence.
386
00:26:03 --> 00:26:09
In the language of matrices,
which was under that board,
387
00:26:09 --> 00:26:13
I could make those the columns
of a matrix.
388
00:26:13 --> 00:26:17
Well, it would be the identity
matrix.
389
00:26:17 --> 00:26:21
Then I would ask,
what's the null space of the
390
00:26:21 --> 00:26:24
identity matrix?
391
00:26:24 --> 00:26:27
And you would say it's only the
zero vector.
392
00:26:27 --> 00:26:31
And I would say,
fine, then the columns are
393
00:26:31 --> 00:26:32.19
independent.
394
00:26:32.19 --> 00:26:36
The only thing -- the identity
times a vector giving zero,
395
00:26:36 --> 00:26:40
the only vector that does that
is zero.
396
00:26:40 --> 00:26:40
OK.
397
00:26:40 --> 00:26:45
Now that's not the only basis.
398
00:26:45 --> 00:26:47
Far from it.
399
00:26:47 --> 00:26:53
Tell me another basis,
a second basis,
400
00:26:53 --> 00:26:55
another basis.
401
00:26:55 --> 00:27:02
So, give me -- well,
I'll just start it out.
402
00:27:02 --> 00:27:04
One one two.
403
00:27:04 --> 00:27:06
Two two five.
404
00:27:06 --> 00:27:12.69
Suppose I stopped there.
405
00:27:12.69 --> 00:27:18
Has that little bunch of
vectors got the properties that
406
00:27:18 --> 00:27:21
I'm asking for in a basis for
R^3?
407
00:27:21 --> 00:27:25
We're looking for a basis for
R^3.
408
00:27:25 --> 00:27:29
Are they independent,
those two column vectors?
409
00:27:29 --> 00:27:30
Yes.
410
00:27:30 --> 00:27:33
Do they span R^3?
411
00:27:33 --> 00:27:33.35
No.
412
00:27:33.35 --> 00:27:34
Our feeling is no.
413
00:27:34 --> 00:27:36
Our feeling is no.
414
00:27:36 --> 00:27:41
Our feeling is that there're
some vectors in R3 that are not
415
00:27:41 --> 00:27:43
combinations of those.
416
00:27:43 --> 00:27:43.29
OK.
417
00:27:43.29 --> 00:27:47
So suppose I add in -- I need
another vector then,
418
00:27:47 --> 00:27:51
because these two don't span
the space.
419
00:27:51 --> 00:27:51
OK.
420
00:27:51 --> 00:27:56
Now it would be foolish for me
to put in three three seven,
421
00:27:56 --> 00:27:58
right, as the third vector.
422
00:27:58 --> 00:28:00
That would be a goof.
423
00:28:00 --> 00:28:04
Because that,
if I put in three three seven,
424
00:28:04 --> 00:28:08.65
those vectors would be
dependent, right?
425
00:28:08.65 --> 00:28:12
If I put in three three seven,
it would be the sum of those
426
00:28:12 --> 00:28:16
two, it would lie in the same
plane as those.
427
00:28:16 --> 00:28:18
It wouldn't be independent.
428
00:28:18 --> 00:28:21.31
My attempt to create a basis
would be dead.
429
00:28:21.31 --> 00:28:25
But if I take -- so what vector
can I take?
430
00:28:25 --> 00:28:29
I can take any vector that's
not in that plane.
431
00:28:29 --> 00:28:33
Let me try -- I hope that 3 3 8
would do it.
432
00:28:33 --> 00:28:37
At least it's not the sum of
those two vectors.
433
00:28:37 --> 00:28:40
But I believe that's a basis.
434
00:28:40 --> 00:28:45
And what's the test then,
for that to be a basis?
435
00:28:45 --> 00:28:51.57
Because I just picked those
numbers, and if I had picked,
436
00:28:51.57 --> 00:28:57
5 7 -14 how would we know do we
have a basis or don't we?
437
00:28:57 --> 00:29:02
You would put them in the
columns of a matrix,
438
00:29:02 --> 00:29:08
and you would do elimination,
row reduction --
439
00:29:08 --> 00:29:15
and you would see do you get
any free variables or are all
440
00:29:15 --> 00:29:18
the columns pivot columns.
441
00:29:18 --> 00:29:25
Well now actually we have a
square -- the matrix would be
442
00:29:25 --> 00:29:27
three by three.
443
00:29:27 --> 00:29:33
So, what's the test on the
matrix then?
444
00:29:33 --> 00:29:42
The matrix -- so in this case,
when my space is R^3 and I have
445
00:29:42 --> 00:29:48
three vectors,
my matrix is square and what I
446
00:29:48 --> 00:29:57
asking about that matrix in
order for those columns to be a
447
00:29:57 --> 00:29:58.34
basis?
448
00:29:58.34 --> 00:30:05
So in this -- for R^n,
if I have -- n vectors give a
449
00:30:05 --> 00:30:12
basis if the n by n matrix with
those columns,
450
00:30:12 --> 00:30:18
with those columns,
is what?
451
00:30:18 --> 00:30:21
What's the requirement on that
matrix?
452
00:30:21 --> 00:30:24
Invertible, right,
right.
453
00:30:24 --> 00:30:27
The matrix should be
invertible.
454
00:30:27 --> 00:30:32
For a square matrix,
that's the, that's the perfect
455
00:30:32 --> 00:30:33
answer.
456
00:30:33 --> 00:30:34
Is invertible.
457
00:30:34 --> 00:30:39
So that's when,
that's when the space is the
458
00:30:39 --> 00:30:42
whole space R^n.
459
00:30:42 --> 00:30:46
Let me, let me be sure you're
with me here.
460
00:30:46 --> 00:30:48
Let me remove that.
461
00:30:48 --> 00:30:54
Are those two vectors a basis
for any space at all?
462
00:30:54 --> 00:31:00
Is there a vector space that
those really are a basis for,
463
00:31:00 --> 00:31:06
those, that pair of vectors,
this guy and this 1,
464
00:31:06 --> 00:31:08
1 1 2 and 2 2 5?
465
00:31:08 --> 00:31:12
Is there a space for which
that's a basis?
466
00:31:12 --> 00:31:12
Sure.
467
00:31:12 --> 00:31:16
They're independent,
so they satisfy the first
468
00:31:16 --> 00:31:22
requirement, so what space shall
I take for them to be a basis
469
00:31:22 --> 00:31:22
of?
470
00:31:22 --> 00:31:26
What spaces will they be a
basis for?
471
00:31:26 --> 00:31:28
The one they span.
472
00:31:28 --> 00:31:30
Their combinations.
473
00:31:30 --> 00:31:31
It's a plane,
right?
474
00:31:31 --> 00:31:34
It'll be a plane inside R^3.
475
00:31:34 --> 00:31:39
So if I take this vector 1 1 2,
say it goes there,
476
00:31:39 --> 00:31:43
and this vector 2 2 5,
say it goes there,
477
00:31:43 --> 00:31:49
those are a basis for --
because they span a plane.
478
00:31:49 --> 00:31:52
And they're a basis for the
plane, because they're
479
00:31:52 --> 00:31:52
independent.
480
00:31:52 --> 00:31:56
If I stick in some third guy,
like 3 3 7, which is in the
481
00:31:56 --> 00:31:58
plane -- suppose I put in,
try to put in 3 3 7,
482
00:31:58 --> 00:32:01.45
then the three vectors would
still span the plane,
483
00:32:01.45 --> 00:32:04
but they wouldn't be a basis
anymore because they're not
484
00:32:04 --> 00:32:06.32
independent anymore.
485
00:32:06.32 --> 00:32:06
OK.
486
00:32:06 --> 00:32:15
So, we're looking at the
question of -- again,
487
00:32:15 --> 00:32:24.81
the case with independent
columns is the case where the
488
00:32:24.81 --> 00:32:32
column vectors span the column
space.
489
00:32:32 --> 00:32:36.93
They're independent,
so they're a basis for the
490
00:32:36.93 --> 00:32:38
column space.
491
00:32:38 --> 00:32:38
OK.
492
00:32:38 --> 00:32:41
So now there's one bit of
intuition.
493
00:32:41 --> 00:32:44
Let me go back to all of R^n.
494
00:32:44 --> 00:32:46
So I -- where I put 3 3 8.
495
00:32:46 --> 00:32:47.18
OK.
496
00:32:47.18 --> 00:32:51
The first message is that the
basis is not unique,
497
00:32:51 --> 00:32:53.48
right.
498
00:32:53.48 --> 00:32:56
There's zillions of bases.
499
00:32:56 --> 00:33:00
I take any invertible three by
three matrix,
500
00:33:00 --> 00:33:03
its columns are a basis for
R^3.
501
00:33:03 --> 00:33:07
The column space is R^3,
and if those,
502
00:33:07 --> 00:33:13
if that matrix is invertible,
those columns are independent,
503
00:33:13 --> 00:33:16
I've got a basis for R^3.
504
00:33:16 --> 00:33:19
So there're many,
many bases.
505
00:33:19 --> 00:33:24
But there is something in
common for all those bases.
506
00:33:24 --> 00:33:31
There's something that this
basis shares with that basis and
507
00:33:31 --> 00:33:33.87
every other basis for R^3.
508
00:33:33.87 --> 00:33:35
And what's that?
509
00:33:35 --> 00:33:41
Well, you saw it coming,
because when I stopped here and
510
00:33:41 --> 00:33:47
asked if that was a basis for
R^3, you said no.
511
00:33:47 --> 00:33:53
And I know that you said no
because you knew there weren't
512
00:33:53 --> 00:33:55
enough vectors there.
513
00:33:55 --> 00:33:59
And the great fact is that
there're many,
514
00:33:59 --> 00:34:04
many bases, but -- let me put
in somebody else,
515
00:34:04 --> 00:34:06
just for variety.
516
00:34:06 --> 00:34:11
There are many,
many bases, but they all have
517
00:34:11 --> 00:34:15
the same number of vectors.
518
00:34:15 --> 00:34:20
If we're talking about the
space R^3, then that number of
519
00:34:20 --> 00:34:22
vectors is three.
520
00:34:22 --> 00:34:28
If we're talking about the
space R^n, then that number of
521
00:34:28 --> 00:34:29
vectors is n.
522
00:34:29 --> 00:34:35
If we're talking about some
other space, the column space of
523
00:34:35 --> 00:34:40
some matrix, or the null space
of some matrix,
524
00:34:40 --> 00:34:47
or some other space that we
haven't even thought of,
525
00:34:47 --> 00:34:52
then that still is true that
every basis -- that there're
526
00:34:52 --> 00:34:58.51
lots of bases but every basis
has the same number of vectors.
527
00:34:58.51 --> 00:35:01.77
Let me write that great fact
down.
528
00:35:01.77 --> 00:35:05
Every basis -- we're given a
space.
529
00:35:05 --> 00:35:06
Given a space.
530
00:35:06 --> 00:35:12
R^3 or R^n or some other column
space of a matrix or the null
531
00:35:12 --> 00:35:18.07
space of a matrix or some other
vector space.
532
00:35:18.07 --> 00:35:26
Then the great fact is that
every basis for this,
533
00:35:26 --> 00:35:33
for the space has the same
number of vectors.
534
00:35:33 --> 00:35:43
If one basis has six vectors,
then every other basis has six
535
00:35:43 --> 00:35:44
vectors.
536
00:35:44 --> 00:35:54
So that number six is telling
me like it's telling me how big
537
00:35:54 --> 00:35:58
is the space.
538
00:35:58 --> 00:36:02
It's telling me how many
vectors do I have to have to
539
00:36:02 --> 00:36:03
have a basis.
540
00:36:03 --> 00:36:05
And of course we're seeing it
this way.
541
00:36:05 --> 00:36:08
That number six,
if we had seven vectors,
542
00:36:08 --> 00:36:10
then we've got too many.
543
00:36:10 --> 00:36:13
If we have five vectors we
haven't got enough.
544
00:36:13 --> 00:36:18
Sixes are like just right for
whatever space that is.
545
00:36:18 --> 00:36:21.47
And what do we call that
number?
546
00:36:21.47 --> 00:36:27
That number is -- now I'm ready
for the last definition today.
547
00:36:27 --> 00:36:30
It's the dimension of that
space.
548
00:36:30 --> 00:36:36
So every basis for a space has
the same number of vectors in
549
00:36:36 --> 00:36:36
it.
550
00:36:36 --> 00:36:42
Not the same vectors,
all sorts of bases --
551
00:36:42 --> 00:36:47.68
but the same number of vectors
is always the same,
552
00:36:47.68 --> 00:36:51
and that number is the
dimension.
553
00:36:51 --> 00:36:53
This is definitional.
554
00:36:53 --> 00:36:58
This number is the dimension of
the space.
555
00:36:58 --> 00:36:58
OK.
556
00:36:58 --> 00:36:59
OK.
557
00:36:59 --> 00:37:01
Let's do some examples.
558
00:37:01 --> 00:37:05.71
Because now we've got
definitions.
559
00:37:05.71 --> 00:37:12
Let me repeat the four things,
the four words that have now
560
00:37:12 --> 00:37:15
got defined.
561
00:37:15 --> 00:37:19
Independence,
that looks at combinations not
562
00:37:19 --> 00:37:19
being zero.
563
00:37:19 --> 00:37:23
Spanning, that looks at all the
combinations.
564
00:37:23 --> 00:37:27
Basis, that's the one that
combines independence and
565
00:37:27 --> 00:37:28
spanning.
566
00:37:28 --> 00:37:33
And now we've got the idea of
the dimension of a space.
567
00:37:33 --> 00:37:40
It's the number of vectors in
any basis, because all bases
568
00:37:40 --> 00:37:42.79
have the same number.
569
00:37:42.79 --> 00:37:43
OK.
570
00:37:43 --> 00:37:45.47
Let's take examples.
571
00:37:45.47 --> 00:37:50
Suppose I take,
my space is -- examples now --
572
00:37:50 --> 00:37:55
space is the,
say, the column space of this
573
00:37:55 --> 00:37:57
matrix.
574
00:37:57 --> 00:38:01
Let me write down a matrix.
1 1 1, 2 1 2,
575
00:38:01 --> 00:38:07
and I'll -- just to make it
clear, I'll take the sum there,
576
00:38:07 --> 00:38:12
3 2 3, and let me take the sum
of all -- oh,
577
00:38:12 --> 00:38:17
let me put in one -- yeah,
I'll put in one one one again.
578
00:38:17 --> 00:38:19
OK.
579
00:38:19 --> 00:38:21
So that's four vectors.
580
00:38:21 --> 00:38:25.74
OK, do they span the column
space of that matrix?
581
00:38:25.74 --> 00:38:29
Let me repeat,
do they span the column space
582
00:38:29 --> 00:38:31.15
of that matrix?
583
00:38:31.15 --> 00:38:31
Yes.
584
00:38:31 --> 00:38:35.72
By definition,
that's what the column space --
585
00:38:35.72 --> 00:38:38
where it comes from.
586
00:38:38 --> 00:38:41
Are they a basis for the column
space?
587
00:38:41 --> 00:38:42
Are they independent?
588
00:38:42 --> 00:38:44
No, they're not independent.
589
00:38:44 --> 00:38:47
There's something in that null
space.
590
00:38:47 --> 00:38:51
Maybe we can -- so let's look
at the null space of the matrix.
591
00:38:51 --> 00:38:56
Tell me a vector that's in the
null space of that matrix.
592
00:38:56 --> 00:39:03
So I'm looking for some vector
that combines those columns and
593
00:39:03 --> 00:39:05
produces the zero column.
594
00:39:05 --> 00:39:11
Or in other words,
I'm looking for solutions to A
595
00:39:11 --> 00:39:12
X equals zero.
596
00:39:12 --> 00:39:17
So tell me a vector in the null
space.
597
00:39:17 --> 00:39:20
Maybe -- well,
this was, this column was that
598
00:39:20 --> 00:39:24
one plus that one,
so maybe if I have one of those
599
00:39:24 --> 00:39:28
and minus one of those that
would be a vector in the null
600
00:39:28 --> 00:39:28
space.
601
00:39:28 --> 00:39:32
So, you've already told me now,
are those vectors independent,
602
00:39:32 --> 00:39:37
the answer is -- those column
vectors, the answer is -- no.
603
00:39:37 --> 00:39:38
Right?
604
00:39:38 --> 00:39:40
They're not independent.
605
00:39:40 --> 00:39:43.33
Because -- you knew they
weren't independent.
606
00:39:43.33 --> 00:39:47
Anyway, minus one of this minus
one of this plus one of this
607
00:39:47 --> 00:39:50.03
zero of that is the zero vector.
608
00:39:50.03 --> 00:39:50
OK.
609
00:39:50 --> 00:39:52
OK, so they're not independent.
610
00:39:52 --> 00:39:56
They span, but they're not
independent.
611
00:39:56 --> 00:39:59
Tell me a basis for that column
space.
612
00:39:59 --> 00:40:02.73
What's a basis for the column
space?
613
00:40:02.73 --> 00:40:07
These are all the questions
that the homework asks,
614
00:40:07 --> 00:40:10
the quizzes ask,
the final exam will ask.
615
00:40:10 --> 00:40:16
Find a basis for the column
space of this matrix.
616
00:40:16 --> 00:40:16
OK.
617
00:40:16 --> 00:40:21
Now there's many answers,
but give me the most natural
618
00:40:21 --> 00:40:21.67
answer.
619
00:40:21.67 --> 00:40:23
Columns one and two.
620
00:40:23 --> 00:40:25
Columns one and two.
621
00:40:25 --> 00:40:27
That's the natural answer.
622
00:40:27 --> 00:40:31
Those are the pivot columns,
because, I mean,
623
00:40:31 --> 00:40:33
we s- we begin systematically.
624
00:40:33 --> 00:40:37
We look at the first column,
it's OK.
625
00:40:37 --> 00:40:40
We can put that in the basis.
626
00:40:40 --> 00:40:43
We look at the second column,
it's OK.
627
00:40:43 --> 00:40:46
We can put that in the basis.
628
00:40:46 --> 00:40:49
The third column we can't put
in the basis.
629
00:40:49 --> 00:40:52
The fourth column we can't,
again.
630
00:40:52 --> 00:40:58
So the rank of the matrix is --
what's the rank of our matrix?
631
00:40:58 --> 00:40:59
Two.
632
00:40:59 --> 00:41:00.03
Two.
633
00:41:00.03 --> 00:41:08
And, and now that rank is also
-- we also have another word.
634
00:41:08 --> 00:41:12
We, we have a great theorem
here.
635
00:41:12 --> 00:41:15
The rank of A,
that rank r,
636
00:41:15 --> 00:41:22
is the number of pivot columns
and it's also -- well,
637
00:41:22 --> 00:41:27
so now please use my new word.
638
00:41:27 --> 00:41:34
This, it's the number two,
of course, two is the rank of
639
00:41:34 --> 00:41:41.13
my matrix,
it's the number of pivot
640
00:41:41.13 --> 00:41:46.9
columns, those pivot columns
form a basis,
641
00:41:46.9 --> 00:41:50
of course, so what's two?
642
00:41:50 --> 00:41:53
It's the dimension.
643
00:41:53 --> 00:41:59
The rank of A,
the number of pivot columns,
644
00:41:59 --> 00:42:04.07
is the dimension of the column
space.
645
00:42:04.07 --> 00:42:06
Of course, you say.
646
00:42:06 --> 00:42:08
It had to be.
647
00:42:08 --> 00:42:11
Right.
648
00:42:11 --> 00:42:15
But just watch,
look for one moment at the,
649
00:42:15 --> 00:42:19
the language,
the way the English words get
650
00:42:19 --> 00:42:20
involved here.
651
00:42:20 --> 00:42:25
I take the rank of a matrix,
the rank of a matrix.
652
00:42:25 --> 00:42:31
It's a number of columns and
it's the dimension of -- not the
653
00:42:31 --> 00:42:37
dimension of the matrix,
that's what I want to say.
654
00:42:37 --> 00:42:41
It's the dimension of a space,
a subspace, the column space.
655
00:42:41 --> 00:42:45
Do you see, I don't take the
dimension of A.
656
00:42:45 --> 00:42:47
That's not what I want.
657
00:42:47 --> 00:42:51.56
I'm looking for the dimension
of the column space of A.
658
00:42:51.56 --> 00:42:56
If you use those words right,
it shows you've got the idea
659
00:42:56 --> 00:42:56
right.
660
00:42:56 --> 00:42:58
Similarly here.
661
00:42:58 --> 00:43:01
I don't talk about the rank of
a subspace.
662
00:43:01 --> 00:43:03
It's a matrix that has a rank.
663
00:43:03 --> 00:43:05
I talk about the rank of a
matrix.
664
00:43:05 --> 00:43:09
And the beauty is that these
definitions just merge so that
665
00:43:09 --> 00:43:13.76
the rank of a matrix is the
dimension of its column space.
666
00:43:13.76 --> 00:43:15
And in this example it's two.
667
00:43:15 --> 00:43:19
And then the further question
is, what's a basis?
668
00:43:19 --> 00:43:22
And the first two columns are a
basis.
669
00:43:22 --> 00:43:24.14
Tell me another basis.
670
00:43:24.14 --> 00:43:26
Another basis for the columns
space.
671
00:43:26 --> 00:43:29
You see I just keep hammering
away.
672
00:43:29 --> 00:43:32
I apologize,
but it's, I have to be sure you
673
00:43:32 --> 00:43:33
have the idea of basis.
674
00:43:33 --> 00:43:37
Tell me another basis for the
column space.
675
00:43:37 --> 00:43:42
Well, you could take columns
one and three.
676
00:43:42 --> 00:43:46
That would be a basis for the
column space.
677
00:43:46 --> 00:43:50
Or columns two and three would
be a basis.
678
00:43:50 --> 00:43:53
Or columns two and four.
679
00:43:53 --> 00:44:00
Or tell me another basis that's
not made out of those columns at
680
00:44:00 --> 00:44:01
all?
681
00:44:01 --> 00:44:05
So -- I guess I'm giving you
infinitely many possibilities,
682
00:44:05 --> 00:44:07
so I can't expect a unanimous
answer here.
683
00:44:07 --> 00:44:10
I'll tell you -- but let's look
at another basis,
684
00:44:10 --> 00:44:11
though.
685
00:44:11 --> 00:44:14
I'll just -- because it's only
one out of zillions,
686
00:44:14 --> 00:44:18
I'm going to put it down and
I'm going to erase it.
687
00:44:18 --> 00:44:23
Another basis for the column
space would be -- let's see.
688
00:44:23 --> 00:44:27
I'll put in some things that
are not there.
689
00:44:27 --> 00:44:31
Say, oh well,
just to make it -- my life
690
00:44:31 --> 00:44:33
easy, 2 2 2.
691
00:44:33 --> 00:44:35
That's in the column space.
692
00:44:35 --> 00:44:39
And, that was sort of obvious.
693
00:44:39 --> 00:44:43
Let me take the sum of those,
say 6 4 6.
694
00:44:43 --> 00:44:47.45
Or the sum of all of the
columns, 7 5 7,
695
00:44:47.45 --> 00:44:48
why not.
696
00:44:48 --> 00:44:50
That's in the column space.
697
00:44:50 --> 00:44:55
Those are independent and I've
got the number right,
698
00:44:55 --> 00:44:57
I've got two.
699
00:44:57 --> 00:45:00
Actually, this is a key point.
700
00:45:00 --> 00:45:06.04
If you know the dimension of
the space you're working with,
701
00:45:06.04 --> 00:45:13
and we know that this column --
we know that the dimension,
702
00:45:13 --> 00:45:18
DIM, the dimension of the
column space is two.
703
00:45:18 --> 00:45:25.55
If you know the dimension,
then -- and we have a couple of
704
00:45:25.55 --> 00:45:32
vectors that are independent,
they'll automatically be a
705
00:45:32 --> 00:45:32
basis.
706
00:45:32 --> 00:45:38
If we've got the number of
vectors right,
707
00:45:38 --> 00:45:42
two vectors in this case,
then if they're independent,
708
00:45:42 --> 00:45:45
they can't help but span the
space.
709
00:45:45 --> 00:45:50.38
Because if they didn't span the
space, there'd be a third guy to
710
00:45:50.38 --> 00:45:54
help span the space,
but it couldn't be independent.
711
00:45:54 --> 00:45:58.25
So, it just has to be
independent if we've got the
712
00:45:58.25 --> 00:46:00
numbers right.
713
00:46:00 --> 00:46:01
And they span.
714
00:46:01 --> 00:46:01
OK.
715
00:46:01 --> 00:46:02
Very good.
716
00:46:02 --> 00:46:05
So you got the dimension of a
space.
717
00:46:05 --> 00:46:09
So this was another basis that
I just invented.
718
00:46:09 --> 00:46:09
OK.
719
00:46:09 --> 00:46:13
Now, now I get to ask about the
null space.
720
00:46:13 --> 00:46:17.82
What's the dimension of the
null space?
721
00:46:17.82 --> 00:46:22
So we, we got a great fact
there, the dimension of the
722
00:46:22 --> 00:46:25
column space is the rank.
723
00:46:25 --> 00:46:29.34
Now I want to ask you about the
null space.
724
00:46:29.34 --> 00:46:35
That's the other part of the
lecture, and it'll go on to the
725
00:46:35 --> 00:46:36
next lecture.
726
00:46:36 --> 00:46:36.54
OK.
727
00:46:36.54 --> 00:46:41
So we know the dimension of the
column space is two,
728
00:46:41 --> 00:46:43
the rank.
729
00:46:43 --> 00:46:45
What about the null space?
730
00:46:45 --> 00:46:48
This is a vector in the null
space.
731
00:46:48 --> 00:46:51
Are there other vectors in the
null space?
732
00:46:51 --> 00:46:51
Yes or no?
733
00:46:51 --> 00:46:52
Yes.
734
00:46:52 --> 00:46:55
So this isn't a basis because
it's doesn't span,
735
00:46:55 --> 00:46:56
right?
736
00:46:56 --> 00:47:01
There's more in the null space
than we've got so far.
737
00:47:01 --> 00:47:03
I need another vector at least.
738
00:47:03 --> 00:47:06
So tell me another vector in
the null space.
739
00:47:06 --> 00:47:10
Well, the natural choice,
the choice you naturally think
740
00:47:10 --> 00:47:14
of is I'm going on to the fourth
column, I'm letting that free
741
00:47:14 --> 00:47:18
variable be a one,
and that free variable be a
742
00:47:18 --> 00:47:22
zero, and I'm asking is that
fourth column a combination of
743
00:47:22 --> 00:47:24
my pivot columns?
744
00:47:24 --> 00:47:25
Yes, it is.
745
00:47:25 --> 00:47:28
And it's -- that will do.
746
00:47:28 --> 00:47:34
So what I've written there are
actually the two special
747
00:47:34 --> 00:47:36
solutions, right?
748
00:47:36 --> 00:47:42
I took the two free variables,
free and free.
749
00:47:42 --> 1.
I gave them the values 1 0 or 0
750
1. --> 00:47:46
751
00:47:46 --> 00:47:48
I figured out the rest.
752
00:47:48 --> 00:47:52
So do you see,
let me just say it in words.
753
00:47:52 --> 00:47:55
This vector,
these vectors in the null space
754
00:47:55 --> 00:47:58
are telling me,
they're telling me the
755
00:47:58 --> 00:48:01
combinations of the columns that
give zero.
756
00:48:01 --> 00:48:07
They're telling me in what way
the, the columns are dependent.
757
00:48:07 --> 00:48:10
That's what the null space is
doing.
758
00:48:10 --> 00:48:11
Have I got enough now?
759
00:48:11 --> 00:48:13
And what's the null space now?
760
00:48:13 --> 00:48:16.35
We have to think about the null
space.
761
00:48:16.35 --> 00:48:19
These are two vectors in the
null space.
762
00:48:19 --> 00:48:20.46
They're independent.
763
00:48:20.46 --> 00:48:23
Are they a basis for the null
space?
764
00:48:23 --> 00:48:26.35
What's the dimension of the
null space?
765
00:48:26.35 --> 00:48:30
You see that those questions
just keep coming up all the
766
00:48:30 --> 00:48:30
time.
767
00:48:30 --> 00:48:33
Are they a basis for the null
space?
768
00:48:33 --> 00:48:37
You can tell me the answer even
though we haven't written out a
769
00:48:37 --> 00:48:38
proof of that.
770
00:48:38 --> 00:48:39
Can you?
771
00:48:39 --> 00:48:40
Yes or no?
772
00:48:40 --> 00:48:45
Do these two special solutions
form a basis for the null space?
773
00:48:45 --> 00:48:48
In other words,
does the null space consist of
774
00:48:48 --> 00:48:50
all combinations of those two
guys?
775
00:48:50 --> 00:48:51
Yes or no?
776
00:48:51 --> 00:48:51.87
Yes.
777
00:48:51.87 --> 00:48:52
Yes.
778
00:48:52 --> 00:48:54.57
The null space is two
dimensional.
779
00:48:54.57 --> 00:48:57
The null space,
the dimension of the null
780
00:48:57 --> 00:49:01
space, is the number of free
variables.
781
00:49:01 --> 00:49:08
So the dimension of the null
space is the number of free
782
00:49:08 --> 00:49:09
variables.
783
00:49:09 --> 00:49:15
And at the last second,
give me the formula.
784
00:49:15 --> 00:49:20
This is then the key formula
that we know.
785
00:49:20 --> 00:49:26
How many free variables are
there in terms of R,
786
00:49:26 --> 00:49:31
the rank, m --
the number of rows,
787
00:49:31 --> 00:49:34
n, the number of columns?
788
00:49:34 --> 00:49:36.09
What do we get?
789
00:49:36.09 --> 00:49:40
We have n columns,
r of them are pivot columns,
790
00:49:40 --> 00:49:46
so n-r is the number of free
columns, free variables.
791
00:49:46 --> 00:49:50
And now it's the dimension of
the null space.
792
00:49:50 --> 00:49:52
OK.
793
00:49:52 --> 00:49:53.6
That's great.
794
00:49:53.6 --> 00:49:57
That's the key spaces,
their bases,
795
00:49:57 --> 00:49:59
and their dimensions.
796
00:49:59 --> 00:50:02
Thanks.