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Okay.
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This is lecture six in linear
algebra, and we're at the start
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of this new chapter,
chapter three in the text,
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which is really getting to the
center of linear algebra.
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And I had time to make a first
start on it at the end of
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lecture five.
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But now is lecture six is
officially the lecture on vector
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spaces and subspaces.
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And then especially -- there
are two subspaces that we're
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specially interested in.
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One is the column space of a
matrix, the other is the null
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space of the matrix.
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So, I got to tell you what
those are.
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Okay.
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So, first to remember from
lecture five,
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what is a vector space?
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It's a bunch of vectors that --
where I'm allowed -- where I can
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add -- I can add any two vectors
in the space and the answer
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stays in the space.
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Or I can multiply any vector in
the space by any constant and
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the result stays in the space.
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So that's -- in fact if I
combine those two into one,
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you can see that -- if I can
add and I can multiply by
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numbers, that really means that
I can take linear combinations.
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So the quick way to say it is
that all linear combinations,
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C --
any multiple of V plus any
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multiple of W stay in the space.
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So, can I give you examples
that are vector spaces and also
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some examples that are not,
to make that point clear?
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So, suppose I'm in three
dimensions.
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Then one way to get us one
space is the whole three
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dimensional space.
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So the whole space R^3,
three dimensional space,
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would be a vector space,
because if I have a couple of
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vectors I can add them and I'm
certainly okay and they follow
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all the rules.
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So R^3 is easy.
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Now I'm interested also in
subspaces.
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So there's this key word,
subspaces.
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That's a space -- that's some
vectors inside the given space,
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inside R three that still make
up a vector space of their own.
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It's a vector space inside a
vector space.
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And the simplest example was a
plane.
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So, like, can I just sketch it
-- there is a plane.
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It's got to go through the
origin, and of course it goes
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infinitely far.
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That's of that's a subspace
now.
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Do you see that if I have two
vectors on the plane and I add
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them, the result stays in the
plane.
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If I take a vector in the plane
and I multiply by minus two,
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I'm still in the plane.
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So that plane is a subspace.
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So let me just make that point.
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Plane through zero,
through that zero zero zero is
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a subspace.
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Okay.
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And also, another subspace
would be a line.
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A line through zero zero zero
-- yeah, the line has to go
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through the origin.
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All subspaces have got to
contain the origin,
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contain zero -- the zero
vector.
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So this line is a subspace.
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Really, if I want to say it
really correctly,
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I should say a subspace of R^3.
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That of R^3 was,
like, understood there.
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Now -- so let me call this
plane P.
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And let me call this line L.
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And let me ask you about other
sets of vectors.
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Suppose I took --
yeah -- so here's a first
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question.
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Suppose I take two subspaces,
like P and L.
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And I just put them together,
take their union,
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take all the vectors -- so now
you've got P and L in mind,
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here.
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So I have two subspaces.
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I have two subspaces and,
for example,
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P -- a plane and L a line.
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Okay.
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Now I want to ask you about the
union of those.
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So P union L.
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This is all vectors in P or L
or both.
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Is that a subspace?
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Is this a subspace?
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This is or is not a subspace?
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Because we're -- I just want to
be sure that I've got the
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central idea.
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Suppose I take the vectors in
the plane and also the vectors
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on that line,
put them together,
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so I've got a bunch of vectors,
is it a subspace?
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Can you give me,
like, so the camera can hear it
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or maybe the tape.
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Can you say yes or no?
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Do I have a subspace if I put
-- if I take all the vectors on
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the plane plus all -- and all
the ones on the line and just
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join them together --
but I'm not taking this guy
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that's -- actually,
I'm not taking most of them,
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because most vectors are not on
the line or the plane,
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they're off somewhere else.
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Do I have a subspace?
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STUDENTS: No.
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STRANG: Right.
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Thank you.
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No.
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Because -- why not?
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Because I can't add.
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Because if I that requirement
isn't satisfied.
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If I take one vector like this
guy and another vector that
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happens to come from L and add,
I'm off somewhere else.
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You see that I've gone outside
the union if I just add
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something from P and something
from L, then normally what'll
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happen is I'm outside the union
-- and I don't have a subspace.
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So the correct answer is -- is
not.
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Okay.
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Now let me ask you about -- the
other thing we do is take the
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intersection.
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So what does intersection mean?
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Intersection means all vectors
that are in both P and L.
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Is this a subspace.
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Yeah, so I guess I want to go
back up to the same question.
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This is or is not a subspace?
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And you can answer me -- answer
the question first for this
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particular example,
this picture I drew.
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What is P intersect L for this
case?
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STUDENT: It's only zero.
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STRANG: It's only zero.
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At least, sort of this was the
artist's idea as he drew it
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that, that that line L was not
in the plane and,
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went off somewhere else --
and then the only point that
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was in common was the zero
vector.
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Is the zero vector by itself a
subspace?
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STUDENT: Yes.
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STRANG: Yes,
absolutely.
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And what about,
if I don't have this plane and
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this line but any subspace and
any other subspace?
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So now -- can I ask that
question for any two subspaces?
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So maybe I'll write it up here.
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If I'm strong enough.
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Okay.
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So this is the general
question.
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I have subspaces,
say S and T.
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And I want to ask you about
their intersection S intersect T
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and I want -- it is a subspace.
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Do you see why?
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Do you see why if I take the
vectors that are in both one-
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th- that are in both of the
subspaces -- so that's like a
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smaller set of vectors,
probably, because it's --
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we've added requirements.
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It has to be in S and in T.
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How do I know that's a
subspace?
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Can we just think through that
abstract stuff and then I get to
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the examples.
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Okay.
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So why?
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Suppose I take a couple of
vectors that are in the
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intersection.
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Why is the sum also in the
intersection?
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Okay, so let me give a name to
these vectors,
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say V and W.
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They're in the intersection.
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So that means they're both in
S.
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Also means they're both in T.
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So what can I say about V plus
W?
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Is it in S?
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Yes.
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Right?
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If I take two vectors,
V and W that are both in S,
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then the sum is in S,
because S was a subspace.
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And if they're both in T and I
add them, then the result is
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also in T, because T was a
subspace.
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So the result V plus W is in
the intersection.
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It's in both and requirement
one is satisfied.
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Requirement two's the same.
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If I take a vector that's in
both, I multiply by seven.
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Seven times that vector is in
S, because the vector was in S.
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Seven times that vector's in T
because the original one was in
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T.
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So seven times that vector is
in the intersection.
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In other words,
when you take the intersection
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of two subspaces,
you get probably a smaller
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subspace, but it is a subspace.
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Okay.
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So that's like sort of just
emphasizing what these two
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requirements mean.
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Again --
Let me circle those,
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because those are so important.
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The sum and the scale of
multiplication which combines
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into linear combinations.
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That's what you have to do
inside the subspace.
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Okay.
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On to the column space.
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Okay.
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So my lecture last time started
that and I want to continue it.
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Okay.
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Column space of a matrix.
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Of A.
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Okay.
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Can I take an example?
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Say one two three four.
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One one one one.
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Two three four five.
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Okay.
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That's my matrix A.
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So, it's got columns,
three columns.
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Those columns are vectors,
so the column space of this A,
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of this A -- let's stay with
this example for a while.
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The column space of this matrix
is a subspace of R -- R what?
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So what space are we in if I'm
looking at the columns of this
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matrix?
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R^4 , right?
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These are vectors in R^4,
they're four dimensional
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vectors.
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So it's this column space of A
is a subspace of R^4 here,
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because A was four by -- A is a
four by three matrix.
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This tells me how many rows
there are, how many components
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in a column, and so we're in
R^4.
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Okay, now what's in that
subspace?
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So the column space of A,
it's a subspace of R^4.
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I call it the column space of
A, like that.
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So that's my little symbol for
some subspace of R^4.
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What's in that subspace?
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Well, that column certainly is.
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One two three four.
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This column is in.
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This column is in,
and what else?
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So it's got the columns of A in
it, but that's not enough,
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certainly.
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Right?
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I don't have a subspace if I
just put in three vectors.
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So how do I fill that out to be
a subspace?
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I take their linear
combinations.
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So the column space of A is all
linear combinations --
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combinations of the columns.
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And that does give me a
subspace.
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It does give me a vector space,
because if I have one linear
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combination and I multiply by
eleven, I've got another linear
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combination.
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If I have a linear combination,
I add to another linear
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combination I get a third
combination.
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So that -- this is like the
smallest space -- like,
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it's got to have those three
columns in it,
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and it has to have their
combinations and that's where we
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stop.
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Okay.
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Now I'm going to be interested
in that space.
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So I, like -- get some idea of
what's in that space.
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How big is that space?
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Is that space the whole four
dimensional space?
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Or is it a subspace inside?
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Can you -- let me just see if
we can get a yes or no answer
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sometimes without being ready
for the complete proof.
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What do you think?
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Is the subspace that I'm
talking about here,
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the combinations of those three
guys, does that fill the full
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four dimensional space?
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Maybe yes or no on that one.
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No.
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No.
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Somehow our feeling is,
and it happens to be right,
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that if we start with three
vectors and take their
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combinations,
we can't get the whole four
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dimensional space.
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Now -- so somehow we get a
smaller space.
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But how much smaller?
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That's going to come up here.
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That's not so immediate.
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Let me first make this critical
connection with --
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with, linear equations,
because behind our abstract
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definition, we have a purpose.
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And that is to understand Ax=b.
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So suppose I make the
connection -- w- w- does A x=b
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always have a solution for every
b?
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Have a solution for every
right-hand side?
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I guess that's going to be a
yes or no question.
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And then I'm going to ask which
right-hand sides are okay?
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That's really the question I'm
after, is which right-hand sides
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(b) do make up -- you can see
from the way I'm speaking what
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the question --
As it is.
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The answer is no.
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A x=b does not have a solution
for every b.
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Why do I say no?
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Because A x=b is -- like,
this is four equations,
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and only three unknowns.
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Right?
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X is -- let me right out that
whole -- what the whole thing
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looks like.
286
00:17:46 --> 00:17:46
Yeah.
287
00:17:46 --> 00:17:50
Let me write out A x=b.
288
00:17:50 --> 00:17:56
A x is -- these columns are one
two three four.
289
00:17:56 --> 00:18:04
One one one one and two three
four five.
290
00:18:04 --> 00:18:08
Then x, of course,
has three components,
291
00:18:08 --> 00:18:10
x1, x2, x3.
292
00:18:10 --> 00:18:16
And I'm trying to get the --
hit the right-hand side,
293
00:18:16 --> 00:18:18
b1,b2,b3 and b4.
294
00:18:18 --> 00:18:23
So my first point is,
I can't always do it.
295
00:18:23 --> 00:18:29
In a way, that just says again
what you told me five minutes
296
00:18:29 --> 00:18:34
ago --
that the combinations of these
297
00:18:34 --> 00:18:37
columns don't fill the whole
four dimensional space.
298
00:18:37 --> 00:18:41
There's going to be some
vectors b, a lot of vectors b,
299
00:18:41 --> 00:18:45
that are not combinations of
these three columns,
300
00:18:45 --> 00:18:48
because the combinations of
those columns are,
301
00:18:48 --> 00:18:53
like, going to be just a little
plane or something inside --
302
00:18:53 --> 00:18:54
inside R^4.
303
00:18:54 --> 00:19:00
Now, so and you see that I do
have four equations and only
304
00:19:00 --> 00:19:01
three unknowns.
305
00:19:01 --> 00:19:06
So, like anybody is going to
say, no you dope,
306
00:19:06 --> 00:19:11
you can't usually solve four
equations with only three
307
00:19:11 --> 00:19:13
unknowns.
308
00:19:13 --> 00:19:16
But now I want to say sometimes
you can.
309
00:19:16 --> 00:19:20
For some right-hand sides,
I can solve this.
310
00:19:20 --> 00:19:25
So that's the bunch of
right-hand sides that I'm
311
00:19:25 --> 00:19:27
interested in right now.
312
00:19:27 --> 00:19:31.45
Which right-hand sides allow me
to solve this?
313
00:19:31.45 --> 00:19:35
This is the question for today.
314
00:19:35 --> 00:19:41
It's going to have,
like, a nice clear answer.
315
00:19:41 --> 00:19:49
So my question is -- is which
bs, which vectors b,
316
00:19:49 --> 00:19:53.7
allow this system to be solved?
317
00:19:53.7 --> 00:19:59
And I want to ask you -- so
that's, like,
318
00:19:59 --> 00:20:07
gets two question marks to
indicate that's --
319
00:20:07 --> 00:20:10.23
this is the important question.
320
00:20:10.23 --> 00:20:14
Okay, first,
before we give a total answer,
321
00:20:14 --> 00:20:16
give me just a partial answer.
322
00:20:16 --> 00:20:21
Tell me one right-hand side
that I know I can solve this
323
00:20:21 --> 00:20:22
thing for.
324
00:20:22 --> 00:20:24
So -- all zeroes.
325
00:20:24 --> 00:20:24
Okay.
326
00:20:24 --> 00:20:28
That's the, like,
guaranteed.
327
00:20:28 --> 00:20:32
If these were all zero,
then I know I can solve it,
328
00:20:32 --> 00:20:36.43
let the x-s all be zero,
no problem.
329
00:20:36.43 --> 00:20:36
Okay.
330
00:20:36 --> 00:20:39
So that's always a -- okay.
331
00:20:39 --> 00:20:42
A x=0 I can always solve.
332
00:20:42 --> 00:20:47
Now tell me another right-hand
side, just a specific set of
333
00:20:47 --> 00:20:53
numbers for which I can solve
these three --
334
00:20:53 --> 00:20:57
these four equations with only
three unknowns,
335
00:20:57 --> 00:21:00
but if you give me a good
right-hand side,
336
00:21:00 --> 00:21:01
I can do it.
337
00:21:01 --> 00:21:03
So tell me one?
338
00:21:03 --> 00:21:04
STUDENT:1 2 3 4.
339
00:21:04 --> 00:21:06
STRANG: 1 2 3 4?
340
00:21:06 --> 00:21:10
If I -- can I solve -- is that
a good right-hand side?
341
00:21:10 --> 00:21:15
Can you solve --
can you find a solution that --
342
00:21:15 --> 00:21:18
X one plus X two plus two X
three is one,
343
00:21:18 --> 00:21:22.9
two X one plus X two plus three
X three is two and two more
344
00:21:22.9 --> 00:21:27.22
equations -- so I'm asking you
to solve in your head in --
345
00:21:27.22 --> 00:21:30
within five seconds,
four equations and three
346
00:21:30 --> 00:21:35
unknowns, but you can do it,
because the right-hand side is,
347
00:21:35 --> 00:21:40
like, showing up here is --
it's one of the columns.
348
00:21:40 --> 00:21:45
So tell me what's the X that
does solve it?
349
00:21:45 --> 00:21:47
One zero zero.
350
00:21:47 --> 00:21:51.89
One zero zero solves it,
because -- well,
351
00:21:51.89 --> 00:21:57
so you can multiply this out by
rows, but oh God,
352
00:21:57 --> 00:22:04
it's much nicer to say --
okay, this is one of this
353
00:22:04 --> 00:22:07
column, zero of this,
zero of this,
354
00:22:07 --> 00:22:12
so it's one of that column,
which is exactly what we
355
00:22:12 --> 00:22:13
wanted.
356
00:22:13 --> 00:22:14
Okay.
357
00:22:14 --> 00:22:17
So there is a b that's okay.
358
00:22:17 --> 00:22:23
Now tell me another B that's
okay, another right-hand side
359
00:22:23 --> 00:22:25
that would be all right?
360
00:22:25 --> 00:22:28
Well -- all ones?
361
00:22:28 --> 00:22:33
Actually -- and then what's the
solution in that case?
362
00:22:33 --> 00:22:34
0 1 0, thanks.
363
00:22:34 --> 00:22:37
And, in fact,
it's much e- like,
364
00:22:37 --> 00:22:42
one way to do it is think of a
solution first,
365
00:22:42 --> 00:22:47
right, and then just see what b
turns out to be.
366
00:22:47 --> 00:22:51.34
What b turns out to be,
right.
367
00:22:51.34 --> 00:22:52
Okay.
368
00:22:52 --> 00:22:58
So I think of a solution -- so
I think of an x,
369
00:22:58 --> 00:23:03
I think of any -- x1,
x2, x3, I do this
370
00:23:03 --> 00:23:09
multiplication and what have I
got?
371
00:23:09 --> 00:23:13
Now I'm ready to answer the big
question.
372
00:23:13 --> 00:23:19
I can solve A x=b exactly when
the right-hand side B is a
373
00:23:19 --> 00:23:22
vector in the column space.
374
00:23:22 --> 00:23:22
Good.
375
00:23:22 --> 00:23:28
I can solve A x=b when b is a
combination of the columns,
376
00:23:28 --> 00:23:34
when it's in the column space
-- so let me write that answer
377
00:23:34 --> 00:23:36
down.
378
00:23:36 --> 00:23:43
I can solve Ax=b exactly when B
is in the column space.
379
00:23:43 --> 00:23:48.67
Let me just say again why that
is.
380
00:23:48.67 --> 00:23:57.33
Because it -- the column space
by its definition contains all
381
00:23:57.33 --> 00:23:59
the combinations.
382
00:23:59 --> 00:24:03
It contains all the Ax-s.
383
00:24:03 --> 00:24:11
The column space really
consists of all vectors A times
384
00:24:11 --> 00:24:13.64
any X.
385
00:24:13.64 --> 00:24:17
So those are the bs that I can
deal with.
386
00:24:17 --> 00:24:23
If b is a combination of the
columns, then that combination
387
00:24:23 --> 00:24:26
tells me what X should be.
388
00:24:26 --> 00:24:32
If b is not a combination of
the columns, then there is no x.
389
00:24:32 --> 00:24:36
There's no way to solve A x
equal b.
390
00:24:36 --> 00:24:37
Okay.
391
00:24:37 --> 00:24:44
So the column space -- that's
really why we're interested in
392
00:24:44 --> 00:24:50
this column space,
because it's the central guy.
393
00:24:50 --> 00:24:56
It says when we can solve,
and that --
394
00:24:56 --> 00:24:59
we got to understand this
column space better.
395
00:24:59 --> 00:25:00
Let's see.
396
00:25:00 --> 00:25:03
Do I want to think -- yeah,
somehow -- oh,
397
00:25:03 --> 00:25:06
well, let's just -- as long as
we've got it here,
398
00:25:06 --> 00:25:10
what do I get for this
particular example?
399
00:25:10 --> 00:25:13.59
If I take combinations of this
and this and this,
400
00:25:13.59 --> 00:25:17
I'll tell you the question
that's in my mind.
401
00:25:17 --> 00:25:23
It's not even proper to use
this word yet,
402
00:25:23 --> 00:25:27
but you'll know what it means.
403
00:25:27 --> 00:25:32.41
Are those three columns
independent?
404
00:25:32.41 --> 00:25:40
If I take the combinations of
the three columns -- does each
405
00:25:40 --> 00:25:46
column contribute something new
or now?
406
00:25:46 --> 00:25:51
So that if I take the
combinations of those three
407
00:25:51 --> 00:25:53
columns, do I,
like, get some three
408
00:25:53 --> 00:25:58
dimensional subspace -- do I
have three vectors that are,
409
00:25:58 --> 00:26:00
like, you know,
independent,
410
00:26:00 --> 00:26:02.68
whatever that means?
411
00:26:02.68 --> 00:26:06
Or do I -- is one of those
columns, like,
412
00:26:06 --> 00:26:11
contributing nothing new --
So that actually only two of
413
00:26:11 --> 00:26:15
the columns would have given the
same column space?
414
00:26:15 --> 00:26:18.55
Yeah -- that's a good way to
ask the question.
415
00:26:18.55 --> 00:26:20
Finally I think of it.
416
00:26:20 --> 00:26:25
Can I throw away any columns --
and have the same column space?
417
00:26:25 --> 00:26:26
STUDENT: Yes.
418
00:26:26 --> 00:26:27
STRANG: Yes.
419
00:26:27 --> 00:26:31
And which one do you suggest I
throw away?
420
00:26:31 --> 00:26:33
STUDENT: Column three -- three.
421
00:26:33 --> 00:26:36.71
STRANG: Well,
three is the natural,
422
00:26:36.71 --> 00:26:38
like, guy to target.
423
00:26:38 --> 00:26:40
So if I -- and why?
424
00:26:40 --> 00:26:44
Because -- what's so bad about
three here?
425
00:26:44 --> 00:26:45
Column three?
426
00:26:45 --> 00:26:47
It's the sum of these,
right?
427
00:26:47 --> 00:26:51
So it's not -- if I'm taking --
if I have combinations of these
428
00:26:51 --> 00:26:55
two and I put in this one,
actually, I don't get anything
429
00:26:55 --> 00:26:55.57
more.
430
00:26:55.57 --> 00:26:59
So later on I will call these
pivot columns.
431
00:26:59 --> 00:27:05
And the third guy will not be a
pivot column in this -- with
432
00:27:05 --> 00:27:06
those numbers.
433
00:27:06 --> 00:27:12
Now actually -- honesty makes
me ask you this question.
434
00:27:12 --> 00:27:16
Could I have thrown away column
one?
435
00:27:16 --> 00:27:17
Yes, I could.
436
00:27:17 --> 00:27:19
I could.
437
00:27:19 --> 00:27:22
So when I say pivot columns,
my convention is,
438
00:27:22 --> 00:27:26
okay, I'll keep the first ones
as long as they're not
439
00:27:26 --> 00:27:26
dependent.
440
00:27:26 --> 00:27:29
So I keep this guy,
he's fine, he's a line.
441
00:27:29 --> 00:27:30
I keep the second guy.
442
00:27:30 --> 00:27:32
It's in a second direction.
443
00:27:32 --> 00:27:35
But the third one,
which is in the same plane as
444
00:27:35 --> 00:27:38.51
the first two gives me nothing
new.
445
00:27:38.51 --> 00:27:44
It's dependent in the language
that we will use and I don't
446
00:27:44 --> 00:27:45
need it.
447
00:27:45 --> 00:27:45
Okay.
448
00:27:45 --> 00:27:51
So I would describe the column
space of this matrix as a two
449
00:27:51 --> 00:27:54
dimensional subspace of R^4.
450
00:27:54 --> 00:27:58
A two dimensional subspace of
R^4.
451
00:27:58 --> 00:27:58
Okay.
452
00:27:58 --> 00:28:05
So you're seeing how these
vector spaces work and you --
453
00:28:05 --> 00:28:09
you're seeing that we -- some
idea of dependence or
454
00:28:09 --> 00:28:12
independence is in our future.
455
00:28:12 --> 00:28:13
Okay.
456
00:28:13 --> 00:28:17
Now I want to speak about
another vector space,
457
00:28:17 --> 00:28:18
the null space.
458
00:28:18 --> 00:28:24
So again I'm getting a little
ahead because it's in section
459
00:28:24 --> 00:28:26
three point two,
but that's okay.
460
00:28:26 --> 00:28:28
All right.
461
00:28:28 --> 00:28:32
Now I'm ready for the null
space.
462
00:28:32 --> 00:28:35
Let me keep the same matrix.
463
00:28:35 --> 00:28:41
And this is going to be a
different -- totally different
464
00:28:41 --> 00:28:42
subspace.
465
00:28:42 --> 00:28:44
Totally different.
466
00:28:44 --> 00:28:45
Okay.
467
00:28:45 --> 00:28:49
Now -- so let me make space for
it.
468
00:28:49 --> 00:28:54
Now --
here comes a completely
469
00:28:54 --> 00:28:59
different subspace,
the null space of A.
470
00:28:59 --> 00:29:00
What's in it?
471
00:29:00 --> 00:29:05
It contains not right-hand
sides b.
472
00:29:05 --> 00:29:07
It contains x-s.
473
00:29:07 --> 00:29:15
It contains all x-s that solve
-- this word null is going to --
474
00:29:15 --> 00:29:21
I mean, that's the key word
here, meaning zero.
475
00:29:21 --> 00:29:28
So this contains --
this is all solutions x,
476
00:29:28 --> 00:29:33
and of course x is our vectors,
x1, x2 and x3,
477
00:29:33 --> 00:29:36
to the equation A x=0.
478
00:29:36 --> 00:29:41
Well, four equations,
because we've got -- so,
479
00:29:41 --> 00:29:44
do you see what I'm doing?
480
00:29:44 --> 00:29:49
I'm now saying,
okay, columns were great,
481
00:29:49 --> 00:29:53
the column space we understood.
482
00:29:53 --> 00:29:56
Now I'm interested in x-s.
483
00:29:56 --> 00:30:02
I'm not -- the only b I'm
interested in now is the b of
484
00:30:02 --> 00:30:03
all zeroes.
485
00:30:03 --> 00:30:06
The right-hand side is now
zeroes.
486
00:30:06 --> 00:30:09
And I'm interested in
solutions.
487
00:30:09 --> 00:30:10
x-s.
488
00:30:10 --> 00:30:15
So t- where is this null space
for this example?
489
00:30:15 --> 00:30:20
These x-s are -- have three
components.
490
00:30:20 --> 00:30:29.58
So the null space is a subspace
-- we still have to show it is a
491
00:30:29.58 --> 00:30:32
subspace -- of R^3.
492
00:30:32 --> 00:30:39
So this is -- and we will show
-- these vectors x,
493
00:30:39 --> 00:30:45
this is in R^3,
where the column space was in
494
00:30:45 --> 00:30:49
R^4 in our example.
495
00:30:49 --> 00:30:54
For an m by n matrix,
this is m and this is n,
496
00:30:54 --> 00:31:00
because the number of columns,
n, tells me how many unknowns,
497
00:31:00 --> 00:31:06
how many x-s multiply those
columns, so it tells me the big
498
00:31:06 --> 00:31:10
space, in this case R three that
I'm in.
499
00:31:10 --> 00:31:15
Now tell me --
why don't we figure out what
500
00:31:15 --> 00:31:20
the null space is for this
example, just by looking at it.
501
00:31:20 --> 00:31:23
I mean, that's the beauty of
small examples,
502
00:31:23 --> 00:31:28.6
that my official way to find
null spaces and column spaces
503
00:31:28.6 --> 00:31:32
and get all the facts straight
would be elimination,
504
00:31:32 --> 00:31:34
and we'll do that.
505
00:31:34 --> 00:31:39
But with a small example,
we can see that --
506
00:31:39 --> 00:31:44
see what's going on without
going through the mechanics of
507
00:31:44 --> 00:31:46
elimination.
508
00:31:46 --> 00:31:51
So this null space -- so I'm
talking about -- again,
509
00:31:51 --> 00:31:55
the null space,
and let me copy again the
510
00:31:55 --> 00:31:56
matrix.
511
00:31:56 --> 00:32:01
One two three four,
one one one one and two three
512
00:32:01 --> 00:32:03
four five.
513
00:32:03 --> 00:32:05
What's in the null space?
514
00:32:05 --> 00:32:09
So I'm taking A times x,
so let me right it again,
515
00:32:09 --> 00:32:13
and I want you to solve those
four equations.
516
00:32:13 --> 00:32:18
In fact, I want you to find all
solutions to those four
517
00:32:18 --> 00:32:19
equations.
518
00:32:19 --> 00:32:23
Well, actually,
just first of all find one.
519
00:32:23 --> 00:32:27
Why should I ask you for all of
them?
520
00:32:27 --> 00:32:31
Tell me one -- well,
tell me one solution that y-
521
00:32:31 --> 00:32:35
you don't even have to look at
the matrix to know one solution
522
00:32:35 --> 00:32:37
to this set of equations.
523
00:32:37 --> 00:32:38
It is zero vector.
524
00:32:38 --> 00:32:43
Whatever that matrix is,
its null space contains zero --
525
00:32:43 --> 00:32:49
because A times the zero vector
sure gives the zero right-hand
526
00:32:49 --> 00:32:49
side.
527
00:32:49 --> 00:32:53
So the null space certainly
contains zero.
528
00:32:53 --> 00:32:57.78
A- so it's got a chance to be a
vector space now,
529
00:32:57.78 --> 00:33:00
and it will turn out it is.
530
00:33:00 --> 00:33:00
Okay.
531
00:33:00 --> 00:33:03
Tell me another solution.
532
00:33:03 --> 00:33:08
So this particular null space
-- and of course I'm going to
533
00:33:08 --> 00:33:13.63
call it N(A) for null space --
this contains-- well we've
534
00:33:13.63 --> 00:33:19
already located the zero vector,
and now you're going to tell me
535
00:33:19 --> 00:33:23
another vector that's in the
null space, another solution,
536
00:33:23 --> 00:33:28
another x, another --
you see what I'm asking you for
537
00:33:28 --> 00:33:31
is a combination of those
columns.
538
00:33:31 --> 00:33:35
That's what I'm always looking
at combinations of columns,
539
00:33:35 --> 00:33:40
but now I'm looking at the
weights, the coefficients in the
540
00:33:40 --> 00:33:41
combination.
541
00:33:41 --> 00:33:45
So tell me a good set of
numbers to put in there.
542
00:33:45 --> 00:33:48
One one -- STUDENTS:
Minus one.
543
00:33:48 --> 00:33:50
STRANG: One one minus one.
544
00:33:50 --> 00:33:51.22
Thanks.
545
00:33:51.22 --> 00:33:52
One one minus one.
546
00:33:52 --> 00:33:55
So there's a vector that's in
it.
547
00:33:55 --> 00:33:55.99
Okay.
548
00:33:55.99 --> 00:33:59
But have I got a subspace at
this point?
549
00:33:59 --> 00:34:01
Certainly not,
right?
550
00:34:01 --> 00:34:05
I've got just a couple of
vectors.
551
00:34:05 --> 00:34:07
No way they make a subspace.
552
00:34:07 --> 00:34:11
Tell me -- actually,
why don't I jump the whole way
553
00:34:11 --> 00:34:12
now?
554
00:34:12 --> 00:34:15
Tell me -- well,
tell me one more solution,
555
00:34:15 --> 00:34:18
one more X that would work.
556
00:34:18 --> 00:34:19
Student: 2 2 -2.
557
00:34:19 --> 00:34:20
STRANG: 2 2 -2?
558
00:34:20 --> 00:34:26
Oh, well, tell me all of them,
that would have been easier.
559
00:34:26 --> 00:34:29
Tell me the whole lot,
now.
560
00:34:29 --> 00:34:32
What is the null space for this
matrix?
561
00:34:32 --> 00:34:37
It's all vectors of the form --
what could this be?
562
00:34:37 --> 00:34:42
It could be one one minus one,
it could be it could be any
563
00:34:42 --> 00:34:48
number C, any number --
the same number again and --
564
00:34:48 --> 00:34:50
STUDENTS: Minus.
565
00:34:50 --> 00:34:51
STRANG: Minus C.
566
00:34:51 --> 00:34:57
In other words -- actually,
any multiple of this guy.
567
00:34:57 --> 00:35:02
Oh, that's the perfect
description, because now the
568
00:35:02 --> 00:35:08
zero vector's automatically
included because C could be
569
00:35:08 --> 00:35:09
zero.
570
00:35:09 --> 00:35:14
The vector I had is included,
because C could be one.
571
00:35:14 --> 00:35:16
But now any vector.
572
00:35:16 --> 00:35:18.69
And that's actually it.
573
00:35:18.69 --> 00:35:21.01
And do I have a subspace?
574
00:35:21.01 --> 00:35:23
And what does it look like?
575
00:35:23 --> 00:35:27
It's in -- how would you
describe this,
576
00:35:27 --> 00:35:30
the null space,
this --
577
00:35:30 --> 00:35:34
all these vectors of this form
C C minus C, like,
578
00:35:34 --> 00:35:36
seven seven minus seven.
579
00:35:36 --> 00:35:39
Minus eleven minus eleven plus
eleven.
580
00:35:39 --> 00:35:41
What have I got here?
581
00:35:41 --> 00:35:46
If -- describe that whole null
space of -- what -- if I drew
582
00:35:46 --> 00:35:47
it, what do I draw?
583
00:35:47 --> 00:35:48.95
A line, right?
584
00:35:48.95 --> 00:35:52
The null space is a line.
585
00:35:52 --> 00:35:57
It's the line through -- in R^3
and the vector one one negative
586
00:35:57 --> 00:36:02
one maybe goes down here,
I don't know where it goes,
587
00:36:02 --> 00:36:04
say, down here.
588
00:36:04 --> 00:36:09
There's the vector one one
negative one that you gave me.
589
00:36:09 --> 00:36:14
And where is the vector C C
negative C?
590
00:36:14 --> 00:36:16
It's on this line.
591
00:36:16 --> 00:36:20
Of course, there's zero zero
zero that we had.
592
00:36:20 --> 00:36:25.05
And what we've got is that
whole -- oops -- that whole
593
00:36:25.05 --> 00:36:28
line, going both ways,
through the origin.
594
00:36:28 --> 00:36:31
The null space is a line in
R^3.
595
00:36:31 --> 00:36:32
Okay.
596
00:36:32 --> 00:36:35
For that example,
we could find all the
597
00:36:35 --> 00:36:41
combinations of the columns that
gave zero at sight.
598
00:36:41 --> 00:36:48
Now, can I just take one more
time, to go back to the
599
00:36:48 --> 00:36:53
definition of subspace,
vector space,
600
00:36:53 --> 00:37:01
and ask you -- how do I know
that the null space is a vector
601
00:37:01 --> 00:37:02
space?
602
00:37:02 --> 00:37:08
How I entitled to use this word
space?
603
00:37:08 --> 00:37:15
I'll never use that word space
without meaning that the two
604
00:37:15 --> 00:37:19
requirements are satisfied.
605
00:37:19 --> 00:37:23
Can we just check that they
are?
606
00:37:23 --> 00:37:29
So I'm going to check that --
can I just continue here?
607
00:37:29 --> 00:37:36
Check that -- that the
solutions to A x=0 always give a
608
00:37:36 --> 00:37:37
subspace.
609
00:37:37 --> 00:37:45
And, of course,
the key word is that= "Space."
610
00:37:45 --> 00:37:48
So what do I have to check?
611
00:37:48 --> 00:37:53
I have to show that if I have
one solution,
612
00:37:53 --> 00:37:58
call it x, and another
solution, call it x*,
613
00:37:58 --> 00:38:02
that their sum is also a
solution, right?
614
00:38:02 --> 00:38:05.44
That's a requirement.
615
00:38:05.44 --> 00:38:11
To use that word space,
I have to say --
616
00:38:11 --> 00:38:18
I have to convince myself that
if A x is zero and also -- and A
617
00:38:18 --> 00:38:26
x* is zero, or maybe I should
have said if A v is zero and A w
618
00:38:26 --> 00:38:30
is zero, then what about v plus
w?
619
00:38:30 --> 00:38:34
Shall I -- let me use those
letters.
620
00:38:34 --> 00:38:41
If A v is zero and A w is zero,
then what --
621
00:38:41 --> 00:38:45
if that and that,
then what's my point here?
622
00:38:45 --> 00:38:48
That A times (v+w) must be
zero.
623
00:38:48 --> 00:38:53
That says that if v is in the
null space and w's in the null
624
00:38:53 --> 00:38:57.91
space, then their sum v+w is in
the null space.
625
00:38:57.91 --> 00:39:02
And of course,
now that I've written it down,
626
00:39:02 --> 00:39:07
it's totally absurd,
ridiculously simple --
627
00:39:07 --> 00:39:12
because matrix multiplication
allows me to separate that out
628
00:39:12 --> 00:39:14
into A v plus A w.
629
00:39:14 --> 00:39:17
I shouldn't say absurdly
simple.
630
00:39:17 --> 00:39:19
That was a dumb thing to say.
631
00:39:19 --> 00:39:24
Could -- we've used,
here, a basic law of matrix
632
00:39:24 --> 00:39:25
multiplication.
633
00:39:25 --> 00:39:32
Actually, we've used it without
proving it, but that's okay.
634
00:39:32 --> 00:39:36
We only live so long,
we just skip that proof.
635
00:39:36 --> 00:39:40.88
I think it's called the
distributive law that I can
636
00:39:40.88 --> 00:39:44
split these -- split this into
two pieces.
637
00:39:44 --> 00:39:49
But now you see the point,
that A v is zero and A w is
638
00:39:49 --> 00:39:54
zero so I have zero plus zero
and I do get zero.
639
00:39:54 --> 00:39:55
It checks.
640
00:39:55 --> 00:39:59
And, similarly,
I have to show that if A v is
641
00:39:59 --> 00:40:04.54
zero, then A times any multiple,
say 12v is also zero.
642
00:40:04.54 --> 00:40:06.67
And how do I know that?
643
00:40:06.67 --> 00:40:11
Because I'm allowed to s- bring
that twelve outside.
644
00:40:11 --> 00:40:16
A number, a scaler can move
outside, so I have twelve A vs,
645
00:40:16 --> 00:40:20
twelve zeroes --
I have zero.
646
00:40:20 --> 00:40:21
Okay.
647
00:40:21 --> 00:40:27
Just to -- it's really critical
to understand the -- oh yeah.
648
00:40:27 --> 00:40:34
Here -- I was going to say,
understand what's the point of
649
00:40:34 --> 00:40:35
a vector space?
650
00:40:35 --> 00:40:42
Let me make that point by
changing the right-hand side.
651
00:40:42 --> 00:40:43
Oops.
652
00:40:43 --> 00:40:43
Okay.
653
00:40:43 --> 00:40:48
Let me change the right-hand
side to one two three four.
654
00:40:48 --> 00:40:49
Oh, okay.
655
00:40:49 --> 00:40:54
Why don't we do all of linear
algebra in one lecture,
656
00:40:54 --> 00:40:56
then we -- okay.
657
00:40:56 --> 00:41:00
I would like to know the
solutions to this equation.
658
00:41:00 --> 00:41:03
For those four equations.
659
00:41:03 --> 00:41:06
So I have four equations.
660
00:41:06 --> 00:41:10
I have only three unknowns,
so if I don't have a pretty
661
00:41:10 --> 00:41:14
special right-hand side there
won't be any solution at all.
662
00:41:14 --> 00:41:17
But that is a very special
right-hand side.
663
00:41:17 --> 00:41:20
And we know that there is a
solution, one zero zero.
664
00:41:20 --> 00:41:22
Were there any more solutions?
665
00:41:22 --> 00:41:25
And did they form a vector
space?
666
00:41:25 --> 00:41:26
Okay.
667
00:41:26 --> 00:41:28
So I'm asking two questions
there.
668
00:41:28 --> 00:41:32
One is, do -- so my right-hand
side now is not zero anymore.
669
00:41:32 --> 00:41:36
I'm not looking at the null
space because I changed from
670
00:41:36 --> 00:41:37
zeroes.
671
00:41:37 --> 00:41:40
So my first question is,
do the solutions,
672
00:41:40 --> 00:41:44
if there are any and there are,
do they form a subspace?
673
00:41:44 --> 00:41:48
Let's answer that question
first.
674
00:41:48 --> 00:41:49
Yes or no.
675
00:41:49 --> 00:41:56.84
Do I get a subspace if I look
at the solutions to -- let me go
676
00:41:56.84 --> 00:41:58
back to x1 x2 x3.
677
00:41:58 --> 00:42:05
I'm looking at all the x-s,
at all those vectors in R^3
678
00:42:05 --> 00:42:08.11
that solve A x -b.
679
00:42:08.11 --> 00:42:18
The only thing I've changed is
b isn't zero anymore.
680
00:42:18 --> 00:42:28
Do the x-s, the solutions,
form a vector space?
681
00:42:28 --> 00:42:32
The solutions to this do not
form a subspace.
682
00:42:32 --> 00:42:35
The solutions don't,
because -- how shall I see
683
00:42:35 --> 00:42:36
that?
684
00:42:36 --> 00:42:40
The zero vector is not a
solution, so I never even got
685
00:42:40 --> 00:42:40
started.
686
00:42:40 --> 00:42:44
The zero vector doesn't solve
this system.
687
00:42:44 --> 00:42:48
I can't -- solutions can't be a
vector space.
688
00:42:48 --> 00:42:50
Now what are they like?
689
00:42:50 --> 00:42:54.82
Well, we'll see this,
but let's do it for this
690
00:42:54.82 --> 00:42:55
example.
691
00:42:55 --> 00:42:58
So one zero zero was a
solution.
692
00:42:58 --> 00:43:00
You saw that right away.
693
00:43:00 --> 00:43:03
Are there any other solutions?
694
00:43:03 --> 00:43:07
Can you tell me a second
solution to this system of
695
00:43:07 --> 00:43:09
equations?
696
00:43:09 --> 00:43:13
STUDENTS: 0 -1 1 STRANG:
0 -1 1.
697
00:43:13 --> 00:43:16
Boy, that's -- 0 -1 1.
698
00:43:16 --> 00:43:17
Yes.
699
00:43:17 --> 00:43:24
Because that says I take minus
this column plus this one and
700
00:43:24 --> 00:43:26
sure enough.
701
00:43:26 --> 00:43:28
That's right.
702
00:43:28 --> 00:43:32
So there are --
there's a bunch of solutions
703
00:43:32 --> 00:43:33
here.
704
00:43:33 --> 00:43:34
But they're not a subspace.
705
00:43:34 --> 00:43:36.35
I'll tell you what it's like.
706
00:43:36.35 --> 00:43:39
It's like a plane that doesn't
go through the origin,
707
00:43:39 --> 00:43:42
or a line that doesn't go
through the origin.
708
00:43:42 --> 00:43:45
Maybe in this case it's a line
that doesn't go through the
709
00:43:45 --> 00:43:48
origin, if I graft the solutions
to A x equal B.
710
00:43:48 --> 00:43:52.3
So you -- I think you've got
the idea.
711
00:43:52.3 --> 00:43:55
Subspaces have to go through
the origin.
712
00:43:55 --> 00:44:00
If I'm looking at x-s,
then they'd better solve Ax=0.
713
00:44:00 --> 00:44:05
In a way I've got -- my two
subspaces that I --
714
00:44:05 --> 00:44:10
talking about today are kind of
the two ways I can tell you what
715
00:44:10 --> 00:44:12.46
a -- about subspace.
716
00:44:12.46 --> 00:44:15
If I want to tell you about the
column space,
717
00:44:15 --> 00:44:20
I tell you a few columns and I
say take their combinations.
718
00:44:20 --> 00:44:22.91
Like I build up this subspace.
719
00:44:22.91 --> 00:44:26.71
I put in a few vectors,
their combinations make a
720
00:44:26.71 --> 00:44:28
subspace.
721
00:44:28 --> 00:44:33
Now, when I went to -- let me
come back to the one that is a
722
00:44:33 --> 00:44:34
subspace here.
723
00:44:34 --> 00:44:39
Here, when I talked about the
null space, I didn't tell you
724
00:44:39 --> 00:44:40
what's in it.
725
00:44:40 --> 00:44:43
We had to figure out what was
in it.
726
00:44:43 --> 00:44:48
What I told you was the
equations that I'm -- that has
727
00:44:48 --> 00:44:50
to be satisfied.
728
00:44:50 --> 00:44:56
You see those -- like,
those are the two natural ways
729
00:44:56 --> 00:44:59.99
to tell you what's in a
subspace.
730
00:44:59.99 --> 00:45:05.88
I can either give you a few
vectors and say fill it out,
731
00:45:05.88 --> 00:45:11
take combinations --
or I can give you a system of
732
00:45:11 --> 00:45:15
equations, the requirements that
the x-s have to satisfy.
733
00:45:15 --> 00:45:20
And both of those ways produce
subspaces and they're the
734
00:45:20 --> 00:45:23.54
important ways to construct
subspaces.
735
00:45:23.54 --> 00:45:26
Okay, so today's lecture
actually got,
736
00:45:26 --> 00:45:32
the essentials of three point
two, the idea of the null space.
737
00:45:32 --> 00:45:37
Now we have to tackle,
Wednesday, the job of how do we
738
00:45:37 --> 00:45:41
actually get hold of that
subspace in an example that's
739
00:45:41 --> 00:45:44
bigger and we can't see it just
by eye.
740
00:45:44 --> 00:45:45
Okay.
741
00:45:45 --> 00:45:46
See you Wednesday.
742
00:45:46 --> 00:45:49
Thanks.