WEBVTT
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Okay.
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This is lecture six
in linear algebra,
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and we're at the start of this
new chapter, chapter three
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in the text, which
is really getting
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to the center of linear algebra.
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And I had time to make
a first start on it
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at the end of lecture five.
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But now is lecture
six is officially
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the lecture on vector
spaces and subspaces.
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And then especially --
there are two subspaces that
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we're specially interested in.
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One is the column
space of a matrix,
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the other is the null
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 --
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where I can add --
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I can add any two
vectors in the space
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and the answer
stays in the space.
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Or I can multiply any vector
in the space by any constant
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and 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 numbers,
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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 multiple of W stay
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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 dimensional
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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 vectors I
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can add them and I'm certainly
okay and they follow all
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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,
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and of course it
goes 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
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and I add 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 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,
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the line has to go
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. 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 --
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yeah -- so here's
a first 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 --
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so now you've got P
and L in mind, here.
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So I have two subspaces.
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I have two subspaces
and, for example, P --
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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
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the central idea.
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Suppose I take the
vectors in the plane
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and also the vectors
on that line,
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put them together, so I've
got a bunch of vectors,
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is it a subspace?
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Can you give me,
like, so the camera
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can hear it 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
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on 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 that's --
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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
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that 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,
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then normally what'll happen
is I'm outside the union --
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and I don't have a subspace.
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So the correct answer is --
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is not.
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Okay.
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Now let me ask you about --
the other thing we do is take
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the 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, went off
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somewhere else -- and then the
only point that was in common
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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 this line
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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
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T and I want --
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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 --
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so that's like a smaller
set of vectors, probably,
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because it's -- 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
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and then I get to the examples.
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Okay.
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So why?
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Suppose I take a
couple of vectors
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that are in the 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, 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,
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then the result is 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. So seven times that vector
is in the intersection.
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In other words, when you
take the intersection of two
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subspaces, you get probably
a smaller subspace,
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but it is a subspace.
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Okay.
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So that's like sort of
just emphasizing what
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these two requirements mean.
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Again -- Let me circle those,
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 --
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let's stay with this
example for a while.
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The column space of this
matrix is a subspace of R --
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R what?
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So what space are
we in if I'm looking
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at the columns of this 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 --
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A is a four by three matrix.
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This tells me how
many rows there are,
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how many components in a
column, and so we're in R^4.
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Okay, now what's
in that subspace?
00:13:20.030 --> 00:13:24.300
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?
00:13:37.480 --> 00:13:40.010
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?
00:13:45.750 --> 00:13:48.480
So it's got the
columns of A in it,
00:13:48.480 --> 00:13:51.671
but that's not
enough, certainly.
00:13:51.671 --> 00:13:52.170
Right?
00:13:52.170 --> 00:13:55.700
I don't have a subspace if
I just put in three vectors.
00:13:55.700 --> 00:13:59.210
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 --
00:14:15.110 --> 00:14:16.710
combinations of the columns.
00:14:19.480 --> 00:14:22.890
And that does give
me a subspace.
00:14:22.890 --> 00:14:24.890
It does give me a vector
space, because if I
00:14:24.890 --> 00:14:28.040
have one linear combination
and I multiply by eleven,
00:14:28.040 --> 00:14:30.680
I've got another
linear combination.
00:14:30.680 --> 00:14:32.190
If I have a linear
combination, I
00:14:32.190 --> 00:14:33.730
add to another
linear combination
00:14:33.730 --> 00:14:35.970
I get a third combination.
00:14:35.970 --> 00:14:40.260
So that -- this is like
the smallest space --
00:14:40.260 --> 00:14:43.040
like, it's got to have
those three columns in it,
00:14:43.040 --> 00:14:45.270
and it has to have
their combinations
00:14:45.270 --> 00:14:47.110
and that's where we stop.
00:14:47.110 --> 00:14:47.680
Okay.
00:14:47.680 --> 00:14:54.120
Now I'm going to be
interested in that space.
00:14:54.120 --> 00:14:56.930
So I, like -- get some idea
of what's in that space.
00:14:56.930 --> 00:14:58.450
How big is that space?
00:14:58.450 --> 00:15:02.880
Is that space the whole
four dimensional space?
00:15:02.880 --> 00:15:05.290
Or is it a subspace inside?
00:15:05.290 --> 00:15:13.870
Can you -- let me just see if
we can get a yes or no answer
00:15:13.870 --> 00:15:19.930
sometimes without being
ready for the complete proof.
00:15:22.660 --> 00:15:23.540
What do you think?
00:15:23.540 --> 00:15:26.590
Is the subspace that
I'm talking about here,
00:15:26.590 --> 00:15:28.700
the combinations of
those three guys,
00:15:28.700 --> 00:15:32.250
does that fill the full
four dimensional space?
00:15:32.250 --> 00:15:34.900
Maybe yes or no on that one.
00:15:34.900 --> 00:15:35.560
No.
00:15:35.560 --> 00:15:36.060
No.
00:15:36.060 --> 00:15:39.640
Somehow our feeling
is, and it happens
00:15:39.640 --> 00:15:42.730
to be right, that if we
start with three vectors
00:15:42.730 --> 00:15:45.660
and take their combinations,
we can't get the whole four
00:15:45.660 --> 00:15:48.680
dimensional space.
00:15:48.680 --> 00:15:52.590
Now -- so somehow we
get a smaller space.
00:15:52.590 --> 00:15:55.010
But how much smaller?
00:15:55.010 --> 00:15:57.280
That's going to come up here.
00:15:57.280 --> 00:16:00.970
That's not so immediate.
00:16:00.970 --> 00:16:07.950
Let me first make this
critical connection with --
00:16:07.950 --> 00:16:15.980
with, linear equations, because
behind our abstract definition,
00:16:15.980 --> 00:16:17.360
we have a purpose.
00:16:17.360 --> 00:16:19.750
And that is to understand Ax=b.
00:16:19.750 --> 00:16:22.950
So suppose I make
the connection --
00:16:22.950 --> 00:16:33.330
w- w- does A x=b always
have a solution for every b?
00:16:33.330 --> 00:16:47.850
Have a solution for
every right-hand side?
00:16:47.850 --> 00:16:49.880
I guess that's going to
be a yes or no question.
00:16:53.220 --> 00:16:59.580
And then I'm going to ask which
right-hand sides are okay?
00:16:59.580 --> 00:17:02.050
That's really the
question I'm after,
00:17:02.050 --> 00:17:08.900
is which right-hand
sides (b) do make up --
00:17:08.900 --> 00:17:12.210
you can see from the way I'm
speaking what the question --
00:17:12.210 --> 00:17:13.839
As it is.
00:17:13.839 --> 00:17:16.020
The answer is no.
00:17:16.020 --> 00:17:21.710
A x=b does not have a
solution for every b.
00:17:21.710 --> 00:17:26.069
Why do I say no?
00:17:26.069 --> 00:17:34.660
Because A x=b is -- like,
this is four equations,
00:17:34.660 --> 00:17:35.940
and only three unknowns.
00:17:39.590 --> 00:17:40.860
Right?
00:17:40.860 --> 00:17:46.057
X is -- let me right
out that whole --
00:17:46.057 --> 00:17:47.390
what the whole thing looks like.
00:17:49.940 --> 00:17:50.600
Yeah.
00:17:50.600 --> 00:17:54.680
Let me write out A x=b.
00:17:54.680 --> 00:17:59.310
A x is --
00:17:59.310 --> 00:18:02.260
these columns are
one two three four.
00:18:02.260 --> 00:18:07.050
One one one one and
two three four five.
00:18:07.050 --> 00:18:12.410
Then x, of course, has three
components, x1, x2, x3.
00:18:12.410 --> 00:18:15.610
And I'm trying to get the --
00:18:15.610 --> 00:18:19.257
hit the right-hand
side, b1,b2,b3 and b4.
00:18:25.430 --> 00:18:27.850
So my first point is,
I can't always do it.
00:18:30.560 --> 00:18:34.140
In a way, that just says again
what you told me five minutes
00:18:34.140 --> 00:18:36.220
ago --
00:18:36.220 --> 00:18:39.540
that the combinations
of these columns
00:18:39.540 --> 00:18:42.630
don't fill the whole
four dimensional space.
00:18:42.630 --> 00:18:46.190
There's going to be some
vectors b, a lot of vectors b,
00:18:46.190 --> 00:18:50.530
that are not combinations
of these three columns,
00:18:50.530 --> 00:18:53.200
because the combinations
of those columns are, like,
00:18:53.200 --> 00:18:56.130
going to be just a little
plane or something inside --
00:18:56.130 --> 00:18:58.050
inside R^4.
00:18:58.050 --> 00:19:03.990
Now, so and you see that I do
have four equations and only
00:19:03.990 --> 00:19:05.210
three unknowns.
00:19:05.210 --> 00:19:09.110
So, like anybody is going
to say, no you dope,
00:19:09.110 --> 00:19:11.650
you can't usually
solve four equations
00:19:11.650 --> 00:19:13.330
with only three unknowns.
00:19:13.330 --> 00:19:17.710
But now I want to say
sometimes you can.
00:19:17.710 --> 00:19:22.330
For some right-hand
sides, I can solve this.
00:19:22.330 --> 00:19:25.900
So that's the bunch
of right-hand sides
00:19:25.900 --> 00:19:28.710
that I'm interested
in right now.
00:19:28.710 --> 00:19:34.860
Which right-hand sides
allow me to solve this?
00:19:34.860 --> 00:19:36.740
This is the question for today.
00:19:36.740 --> 00:19:40.240
It's going to have, like,
a nice clear answer.
00:19:40.240 --> 00:19:50.800
So my question is -- is
which bs, which vectors b,
00:19:50.800 --> 00:19:55.200
allow this system to be solved?
00:19:59.640 --> 00:20:03.150
And I want to ask you --
00:20:03.150 --> 00:20:08.090
so that's, like, gets two
question marks to indicate
00:20:08.090 --> 00:20:10.350
that's -- this is the
important question.
00:20:10.350 --> 00:20:15.030
Okay, first, before we
give a total answer,
00:20:15.030 --> 00:20:18.000
give me just a partial answer.
00:20:18.000 --> 00:20:20.910
Tell me one right-hand
side that I know
00:20:20.910 --> 00:20:23.160
I can solve this thing for.
00:20:23.160 --> 00:20:24.910
So -- all zeroes.
00:20:24.910 --> 00:20:25.410
Okay.
00:20:25.410 --> 00:20:28.000
That's the, like, guaranteed.
00:20:28.000 --> 00:20:31.990
If these were all zero,
then I know I can solve it,
00:20:31.990 --> 00:20:35.550
let the x-s all be
zero, no problem.
00:20:35.550 --> 00:20:38.890
So that's always a -- okay.
00:20:38.890 --> 00:20:39.660
Okay.
00:20:39.660 --> 00:20:42.460
A x=0 I can always solve.
00:20:42.460 --> 00:20:45.040
Now tell me another
right-hand side,
00:20:45.040 --> 00:20:52.740
just a specific set of numbers
for which I can solve these
00:20:52.740 --> 00:20:55.700
three -- these four equations
with only three unknowns,
00:20:55.700 --> 00:21:00.130
but if you give me a good
right-hand side, I can do it.
00:21:00.130 --> 00:21:00.842
So tell me one?
00:21:00.842 --> 00:21:01.550
STUDENT: 1 2 3 4.
00:21:01.550 --> 00:21:04.740
STRANG: 1 2 3 4?
00:21:04.740 --> 00:21:09.575
If I -- can I solve -- is
that a good right-hand side?
00:21:12.340 --> 00:21:15.340
Can you solve -- can you
find a solution that --
00:21:15.340 --> 00:21:18.630
X one plus X two plus
two X three is one,
00:21:18.630 --> 00:21:22.630
two X one plus X two plus three
X three is two and two more
00:21:22.630 --> 00:21:23.858
equations --
00:21:27.210 --> 00:21:30.220
so I'm asking you to
solve in your head in --
00:21:30.220 --> 00:21:35.750
within five seconds, four
equations and three unknowns,
00:21:35.750 --> 00:21:41.310
but you can do it, because
the right-hand side is, like,
00:21:41.310 --> 00:21:44.610
showing up here is --
it's one of the columns.
00:21:44.610 --> 00:21:48.330
So tell me what's the
X that does solve it?
00:21:48.330 --> 00:21:50.140
One zero zero.
00:21:50.140 --> 00:21:55.960
One zero zero solves
it, because --
00:21:55.960 --> 00:22:01.450
well, so you can multiply
this out by rows, but oh God,
00:22:01.450 --> 00:22:06.360
it's much nicer to say -- okay,
this is one of this column,
00:22:06.360 --> 00:22:09.190
zero of this, zero of this,
so it's one of that column,
00:22:09.190 --> 00:22:11.730
which is exactly what we wanted.
00:22:11.730 --> 00:22:13.290
Okay.
00:22:13.290 --> 00:22:17.280
So there is a b that's okay.
00:22:17.280 --> 00:22:19.250
Now tell me another
B that's okay,
00:22:19.250 --> 00:22:23.030
another right-hand side
that would be all right?
00:22:23.030 --> 00:22:25.055
Well -- all ones?
00:22:29.480 --> 00:22:33.450
Actually -- and then what's
the solution in that case?
00:22:33.450 --> 00:22:36.080
0 1 0, thanks.
00:22:36.080 --> 00:22:40.190
And, in fact, it's
much e- like, one way
00:22:40.190 --> 00:22:44.230
to do it is think of a
solution first, right,
00:22:44.230 --> 00:22:50.380
and then just see what
b turns out to be.
00:22:50.380 --> 00:22:53.030
What b turns out to be, right.
00:22:53.030 --> 00:22:54.790
Okay.
00:22:54.790 --> 00:22:58.220
So I think of a solution
-- so I think of an x,
00:22:58.220 --> 00:23:00.380
I think of any --
00:23:00.380 --> 00:23:03.810
x1, x2, x3, I do
this multiplication
00:23:03.810 --> 00:23:04.980
and what have I got?
00:23:09.640 --> 00:23:13.920
Now I'm ready to answer
the big question.
00:23:13.920 --> 00:23:20.830
I can solve A x=b exactly
when the right-hand side B is
00:23:20.830 --> 00:23:24.920
a vector in the column space.
00:23:24.920 --> 00:23:25.900
Good.
00:23:25.900 --> 00:23:33.140
I can solve A x=b when b is
a combination of the columns,
00:23:33.140 --> 00:23:36.130
when it's in the column space --
00:23:36.130 --> 00:23:39.730
so let me write
that answer down.
00:23:39.730 --> 00:23:57.910
I can solve Ax=b exactly when
B is in the column space.
00:23:57.910 --> 00:24:02.210
Let me just say
again why that is.
00:24:02.210 --> 00:24:07.030
Because it -- the column space
by its definition contains all
00:24:07.030 --> 00:24:07.870
the combinations.
00:24:07.870 --> 00:24:10.710
It contains all the Ax-s.
00:24:10.710 --> 00:24:17.360
The column space really consists
of all vectors A times any X.
00:24:17.360 --> 00:24:22.900
So those are the bs
that I can deal with.
00:24:22.900 --> 00:24:26.130
If b is a combination
of the columns,
00:24:26.130 --> 00:24:32.080
then that combination
tells me what X should be.
00:24:32.080 --> 00:24:35.630
If b is not a combination of
the columns, then there is no x.
00:24:35.630 --> 00:24:38.650
There's no way to
solve A x equal b.
00:24:38.650 --> 00:24:40.420
Okay.
00:24:40.420 --> 00:24:42.700
So the column space --
00:24:42.700 --> 00:24:45.510
that's really why we're
interested in this column
00:24:45.510 --> 00:24:48.170
space, because it's
the central guy.
00:24:48.170 --> 00:24:56.390
It says when we can
solve, and that --
00:24:56.390 --> 00:24:58.880
we got to understand
this column space better.
00:25:02.420 --> 00:25:03.000
Let's see.
00:25:03.000 --> 00:25:07.260
Do I want to think --
00:25:07.260 --> 00:25:10.600
yeah, somehow -- oh,
well, let's just --
00:25:10.600 --> 00:25:13.360
as long as we've got it
here, what do I get for this
00:25:13.360 --> 00:25:14.550
particular example?
00:25:14.550 --> 00:25:24.670
If I take combinations of
this and this and this,
00:25:24.670 --> 00:25:28.230
I'll tell you the question
that's in my mind.
00:25:28.230 --> 00:25:30.930
It's not even proper
to use this word yet,
00:25:30.930 --> 00:25:33.280
but you'll know what it means.
00:25:33.280 --> 00:25:38.060
Are those three
columns independent?
00:25:38.060 --> 00:25:44.370
If I take the combinations
of the three columns --
00:25:44.370 --> 00:25:50.550
does each column contribute
something new or now?
00:25:50.550 --> 00:25:53.360
So that if I take the
combinations of those three
00:25:53.360 --> 00:25:57.160
columns, do I, like, get some
three dimensional subspace --
00:25:57.160 --> 00:26:00.630
do I have three vectors
that are, like, you know,
00:26:00.630 --> 00:26:07.440
independent,
whatever that means?
00:26:07.440 --> 00:26:10.080
Or do I -- is one of
those columns, like,
00:26:10.080 --> 00:26:12.370
contributing nothing new --
00:26:12.370 --> 00:26:15.630
So that actually only
two of the columns
00:26:15.630 --> 00:26:19.040
would have given the
same column space?
00:26:19.040 --> 00:26:21.080
Yeah -- that's a good
way to ask the question.
00:26:21.080 --> 00:26:22.550
Finally I think of it.
00:26:22.550 --> 00:26:25.340
Can I throw away any columns --
00:26:25.340 --> 00:26:27.339
and have the same column space?
00:26:27.339 --> 00:26:27.880
STUDENT: Yes.
00:26:27.880 --> 00:26:29.100
STRANG: Yes.
00:26:29.100 --> 00:26:31.290
And which one do you
suggest I throw away?
00:26:31.290 --> 00:26:32.670
STUDENT: Column three -- three.
00:26:32.670 --> 00:26:37.350
STRANG: Well, three is the
natural, like, guy to target.
00:26:37.350 --> 00:26:39.490
So if I -- and why?
00:26:39.490 --> 00:26:44.750
Because -- what's so
bad about three here?
00:26:44.750 --> 00:26:46.240
Column three?
00:26:46.240 --> 00:26:48.310
It's the sum of these, right?
00:26:48.310 --> 00:26:52.370
So it's not -- if I'm taking --
if I have combinations of these
00:26:52.370 --> 00:26:54.510
two and I put in
this one, actually,
00:26:54.510 --> 00:26:56.780
I don't get anything more.
00:26:56.780 --> 00:27:02.460
So later on I will call
these pivot columns.
00:27:02.460 --> 00:27:06.780
And the third guy will not
be a pivot column in this --
00:27:06.780 --> 00:27:08.760
with those numbers.
00:27:08.760 --> 00:27:12.940
Now actually -- honesty makes
me ask you this question.
00:27:12.940 --> 00:27:16.350
Could I have thrown
away column one?
00:27:16.350 --> 00:27:17.940
Yes, I could.
00:27:17.940 --> 00:27:19.920
I could.
00:27:19.920 --> 00:27:22.910
So when I say pivot
columns, my convention
00:27:22.910 --> 00:27:26.710
is, okay, I'll keep the first
ones as long as they're not
00:27:26.710 --> 00:27:27.590
dependent.
00:27:27.590 --> 00:27:30.930
So I keep this guy,
he's fine, he's a line.
00:27:30.930 --> 00:27:32.100
I keep the second guy.
00:27:32.100 --> 00:27:33.810
It's in a second direction.
00:27:33.810 --> 00:27:39.220
But the third one, which is in
the same plane as the first two
00:27:39.220 --> 00:27:40.790
gives me nothing new.
00:27:40.790 --> 00:27:44.560
It's dependent in the
language that we will use
00:27:44.560 --> 00:27:46.990
and I don't need it.
00:27:46.990 --> 00:27:48.160
Okay.
00:27:48.160 --> 00:27:53.570
So I would describe the column
space of this matrix as a two
00:27:53.570 --> 00:27:56.040
dimensional subspace of R^4.
00:27:58.680 --> 00:28:01.070
A two dimensional
subspace of R^4.
00:28:01.070 --> 00:28:01.990
Okay.
00:28:01.990 --> 00:28:05.430
So you're seeing how these
vector spaces work and you --
00:28:05.430 --> 00:28:08.760
you're seeing that we --
some idea of dependence
00:28:08.760 --> 00:28:11.780
or independence
is in our future.
00:28:11.780 --> 00:28:12.400
Okay.
00:28:12.400 --> 00:28:16.590
Now I want to speak
about another vector
00:28:16.590 --> 00:28:20.460
space, the null space.
00:28:20.460 --> 00:28:23.890
So again I'm getting
a little ahead
00:28:23.890 --> 00:28:28.191
because it's in section three
point two, but that's okay.
00:28:28.191 --> 00:28:28.690
All right.
00:28:28.690 --> 00:28:31.870
Now I'm ready for
the null space.
00:28:31.870 --> 00:28:33.340
Let me keep the same matrix.
00:28:36.830 --> 00:28:39.370
And this is going
to be a different --
00:28:39.370 --> 00:28:41.500
totally different subspace.
00:28:41.500 --> 00:28:43.270
Totally different.
00:28:43.270 --> 00:28:44.930
Okay.
00:28:44.930 --> 00:28:47.370
Now -- so let me
make space for it.
00:28:47.370 --> 00:28:50.190
Now -- here comes a
completely different subspace,
00:28:50.190 --> 00:29:00.120
the null space of A.
00:29:00.120 --> 00:29:00.935
What's in it?
00:29:04.240 --> 00:29:10.070
It contains not
right-hand sides b.
00:29:10.070 --> 00:29:12.750
It contains x-s.
00:29:12.750 --> 00:29:16.570
It contains all
x-s that solve --
00:29:16.570 --> 00:29:18.040
this word null is going to --
00:29:18.040 --> 00:29:22.580
I mean, that's the key
word here, meaning zero.
00:29:22.580 --> 00:29:30.670
So this contains --
this is all solutions x,
00:29:30.670 --> 00:29:36.450
and of course x is our
vectors, x1, x2 and x3,
00:29:36.450 --> 00:29:41.220
to the equation A x=0.
00:29:43.800 --> 00:29:47.800
Well, four equations,
because we've got --
00:29:47.800 --> 00:29:50.230
so, do you see what I'm doing?
00:29:50.230 --> 00:29:53.890
I'm now saying, okay,
columns were great,
00:29:53.890 --> 00:29:56.010
the column space we understood.
00:29:56.010 --> 00:29:59.260
Now I'm interested in x-s.
00:29:59.260 --> 00:30:02.860
I'm not -- the only b I'm
interested in now is the b
00:30:02.860 --> 00:30:03.670
of all zeroes.
00:30:03.670 --> 00:30:06.300
The right-hand
side is now zeroes.
00:30:06.300 --> 00:30:08.550
And I'm interested in solutions.
00:30:12.140 --> 00:30:12.810
x-s.
00:30:12.810 --> 00:30:18.740
So t- where is this null
space for this example?
00:30:18.740 --> 00:30:23.600
These x-s are -- have
three components.
00:30:23.600 --> 00:30:26.310
So the null space
is a subspace --
00:30:26.310 --> 00:30:31.390
we still have to show it
is a subspace -- of R^3.
00:30:31.390 --> 00:30:35.590
So this is -- and
we will show --
00:30:35.590 --> 00:30:46.220
these vectors x, this is in
R^3, where the column space was
00:30:46.220 --> 00:30:50.910
in R^4 in our example.
00:30:50.910 --> 00:30:58.340
For an m by n matrix,
this is m and this is n,
00:30:58.340 --> 00:31:00.570
because the number
of columns, n,
00:31:00.570 --> 00:31:03.000
tells me how many
unknowns, how many x-s
00:31:03.000 --> 00:31:05.950
multiply those
columns, so it tells me
00:31:05.950 --> 00:31:10.480
the big space, in this
case R three that I'm in.
00:31:10.480 --> 00:31:14.300
Now tell me -- why don't we
figure out what the null space
00:31:14.300 --> 00:31:20.310
is for this example,
just by looking at it.
00:31:20.310 --> 00:31:24.740
I mean, that's the
beauty of small examples,
00:31:24.740 --> 00:31:30.500
that my official way to find
null spaces and column spaces
00:31:30.500 --> 00:31:36.200
and get all the facts
straight would be elimination,
00:31:36.200 --> 00:31:37.340
and we'll do that.
00:31:37.340 --> 00:31:40.120
But with a small example,
we can see that --
00:31:40.120 --> 00:31:43.150
see what's going on without
going through the mechanics
00:31:43.150 --> 00:31:43.950
of elimination.
00:31:43.950 --> 00:31:48.670
So this null space --
00:31:48.670 --> 00:31:51.540
so I'm talking about --
again, the null space,
00:31:51.540 --> 00:31:53.520
and let me copy
again the matrix.
00:31:56.530 --> 00:32:04.190
One two three four, one one one
one and two three four five.
00:32:04.190 --> 00:32:05.330
What's in the null space?
00:32:05.330 --> 00:32:11.410
So I'm taking A times x,
so let me right it again,
00:32:11.410 --> 00:32:16.015
and I want you to solve
those four equations.
00:32:19.500 --> 00:32:21.490
In fact, I want you
to find all solutions
00:32:21.490 --> 00:32:24.350
to those four equations.
00:32:24.350 --> 00:32:26.870
Well, actually, just
first of all find one.
00:32:26.870 --> 00:32:28.420
Why should I ask
you for all of them?
00:32:28.420 --> 00:32:31.020
Tell me one -- well, tell
me one solution that y-
00:32:31.020 --> 00:32:35.660
you don't even have to look at
the matrix to know one solution
00:32:35.660 --> 00:32:38.030
to this set of equations.
00:32:38.030 --> 00:32:41.430
It is zero vector.
00:32:41.430 --> 00:32:46.810
Whatever that matrix is, its
null space contains zero --
00:32:46.810 --> 00:32:51.020
because A times the zero vector
sure gives the zero right-hand
00:32:51.020 --> 00:32:51.660
side.
00:32:51.660 --> 00:32:55.700
So the null space
certainly contains zero.
00:32:55.700 --> 00:32:58.220
A- so it's got a chance
to be a vector space now,
00:32:58.220 --> 00:33:00.200
and it will turn out it is.
00:33:00.200 --> 00:33:00.880
Okay.
00:33:00.880 --> 00:33:04.180
Tell me another solution.
00:33:04.180 --> 00:33:07.800
So this particular null space --
and of course I'm going to call
00:33:07.800 --> 00:33:11.110
it N(A) for null space --
00:33:11.110 --> 00:33:17.160
this contains-- well we've
already located the zero
00:33:17.160 --> 00:33:20.970
vector, and now you're going
to tell me another vector
00:33:20.970 --> 00:33:24.550
that's in the null space,
another solution, another x,
00:33:24.550 --> 00:33:26.340
another --
00:33:26.340 --> 00:33:28.140
you see what I'm
asking you for is
00:33:28.140 --> 00:33:29.950
a combination of those columns.
00:33:29.950 --> 00:33:33.150
That's what I'm always looking
at combinations of columns,
00:33:33.150 --> 00:33:40.800
but now I'm looking at the
weights, the coefficients
00:33:40.800 --> 00:33:41.710
in the combination.
00:33:41.710 --> 00:33:46.620
So tell me a good set of
numbers to put in there.
00:33:46.620 --> 00:33:50.420
One one -- STUDENTS: Minus one.
00:33:50.420 --> 00:33:51.530
STRANG: One one minus one.
00:33:51.530 --> 00:33:53.340
Thanks.
00:33:53.340 --> 00:33:55.280
One one minus one.
00:33:55.280 --> 00:33:56.730
So there's a vector
that's in it.
00:33:59.260 --> 00:34:00.250
Okay.
00:34:00.250 --> 00:34:02.600
But have I got a
subspace at this point?
00:34:02.600 --> 00:34:04.690
Certainly not, right?
00:34:04.690 --> 00:34:06.850
I've got just a
couple of vectors.
00:34:06.850 --> 00:34:08.860
No way they make a subspace.
00:34:08.860 --> 00:34:12.940
Tell me -- actually, why don't
I jump the whole way now?
00:34:12.940 --> 00:34:16.840
Tell me -- well, tell
me one more solution,
00:34:16.840 --> 00:34:19.460
one more X that would work.
00:34:19.460 --> 00:34:21.199
Student: 2 2 -2.
00:34:21.199 --> 00:34:22.520
STRANG: 2 2 -2?
00:34:22.520 --> 00:34:27.750
Oh, well, tell me all of them,
that would have been easier.
00:34:27.750 --> 00:34:30.370
Tell me the whole lot, now.
00:34:30.370 --> 00:34:34.389
What is the null
space for this matrix?
00:34:34.389 --> 00:34:38.670
It's all vectors of the
form -- what could this be?
00:34:38.670 --> 00:34:44.900
It could be one one minus one,
it could be it could be any
00:34:44.900 --> 00:34:50.880
number C, any number -- the
same number again and --
00:34:50.880 --> 00:34:52.010
STUDENTS: Minus.
00:34:52.010 --> 00:34:53.420
STRANG: Minus C.
00:34:53.420 --> 00:34:55.010
In other words --
00:34:55.010 --> 00:34:59.910
actually, any
multiple of this guy.
00:34:59.910 --> 00:35:02.270
Oh, that's the
perfect description,
00:35:02.270 --> 00:35:08.050
because now the zero vector's
automatically included
00:35:08.050 --> 00:35:10.040
because C could be zero.
00:35:10.040 --> 00:35:12.810
The vector I had is included,
because C could be one.
00:35:12.810 --> 00:35:14.750
But now any vector.
00:35:14.750 --> 00:35:16.860
And that's actually it.
00:35:20.830 --> 00:35:24.450
And do I have a subspace?
00:35:24.450 --> 00:35:26.060
And what does it look like?
00:35:26.060 --> 00:35:30.620
It's in -- how would you
describe this, the null space,
00:35:30.620 --> 00:35:35.810
this -- all these vectors of
this form C C minus C, like,
00:35:35.810 --> 00:35:38.730
seven seven minus seven.
00:35:38.730 --> 00:35:41.080
Minus eleven minus
eleven plus eleven.
00:35:41.080 --> 00:35:43.810
What have I got here?
00:35:43.810 --> 00:35:46.120
If -- describe that whole
null space of -- what --
00:35:46.120 --> 00:35:50.550
if I drew it, what do I draw?
00:35:50.550 --> 00:35:51.840
A line, right?
00:35:51.840 --> 00:35:53.350
The null space is a line.
00:35:53.350 --> 00:36:02.724
It's the line through -- in R^3
and the vector one one negative
00:36:02.724 --> 00:36:05.140
one maybe goes down here, I
don't know where it goes, say,
00:36:05.140 --> 00:36:06.530
down here.
00:36:06.530 --> 00:36:12.660
There's the vector one one
negative one that you gave me.
00:36:12.660 --> 00:36:15.610
And where is the
vector C C negative C?
00:36:15.610 --> 00:36:17.050
It's on this line.
00:36:17.050 --> 00:36:19.800
Of course, there's zero
zero zero that we had.
00:36:19.800 --> 00:36:23.720
And what we've got is that whole
-- oops -- that whole line,
00:36:23.720 --> 00:36:28.640
going both ways,
through the origin.
00:36:28.640 --> 00:36:31.030
The null space is a line in R^3.
00:36:35.970 --> 00:36:37.930
Okay.
00:36:37.930 --> 00:36:43.040
For that example, we could
find all the combinations
00:36:43.040 --> 00:36:46.590
of the columns that
gave zero at sight.
00:36:46.590 --> 00:36:51.600
Now, can I just
take one more time,
00:36:51.600 --> 00:36:57.860
to go back to the definition
of subspace, vector space,
00:36:57.860 --> 00:37:01.060
and ask you --
00:37:01.060 --> 00:37:05.400
how do I know that the null
space is a vector space?
00:37:05.400 --> 00:37:08.610
How I entitled to
use this word space?
00:37:08.610 --> 00:37:13.070
I'll never use that word space
without meaning that the two
00:37:13.070 --> 00:37:15.800
requirements are satisfied.
00:37:15.800 --> 00:37:18.250
Can we just check that they are?
00:37:18.250 --> 00:37:20.810
So I'm going to check that --
00:37:20.810 --> 00:37:22.660
can I just continue here?
00:37:22.660 --> 00:37:39.240
Check that -- that the
solutions to A x=0 always give
00:37:39.240 --> 00:37:42.740
a subspace.
00:37:42.740 --> 00:37:47.700
And, of course, the key
word is that= "Space."
00:37:47.700 --> 00:37:49.840
So what do I have to check?
00:37:53.410 --> 00:37:57.340
I have to show that if I
have one solution, call it x,
00:37:57.340 --> 00:38:01.240
and another
solution, call it x*,
00:38:01.240 --> 00:38:05.970
that their sum is also
a solution, right?
00:38:05.970 --> 00:38:07.290
That's a requirement.
00:38:07.290 --> 00:38:10.010
To use that word
space, I have to say --
00:38:10.010 --> 00:38:18.650
I have to convince myself that
if A x is zero and also --
00:38:18.650 --> 00:38:25.140
and A x* is zero, or maybe I
should have said if A v is zero
00:38:25.140 --> 00:38:30.130
and A w is zero, then
what about v plus w?
00:38:30.130 --> 00:38:32.630
Shall I -- let me
use those letters.
00:38:32.630 --> 00:38:47.300
If A v is zero and A w is zero,
then what -- if that and that,
00:38:47.300 --> 00:38:49.240
then what's my point here?
00:38:49.240 --> 00:38:55.170
That A times (v+w) must be zero.
00:38:55.170 --> 00:38:59.590
That says that if v is in the
null space and w's in the null
00:38:59.590 --> 00:39:03.680
space, then their sum
v+w is in the null space.
00:39:03.680 --> 00:39:06.250
And of course, now that
I've written it down,
00:39:06.250 --> 00:39:11.460
it's totally absurd,
ridiculously simple --
00:39:11.460 --> 00:39:17.290
because matrix multiplication
allows me to separate that out
00:39:17.290 --> 00:39:19.460
into A v plus A w.
00:39:22.050 --> 00:39:23.560
I shouldn't say absurdly simple.
00:39:23.560 --> 00:39:24.840
That was a dumb thing to say.
00:39:24.840 --> 00:39:28.960
Could -- we've used, here,
a basic law of matrix
00:39:28.960 --> 00:39:30.580
multiplication.
00:39:30.580 --> 00:39:34.070
Actually, we've used it without
proving it, but that's okay.
00:39:34.070 --> 00:39:39.340
We only live so long,
we just skip that proof.
00:39:39.340 --> 00:39:42.820
I think it's called the
distributive law that I can
00:39:42.820 --> 00:39:45.770
split these -- split
this into two pieces.
00:39:45.770 --> 00:39:52.150
But now you see the point, that
A v is zero and A w is zero
00:39:52.150 --> 00:39:54.520
so I have zero plus
zero and I do get zero.
00:39:54.520 --> 00:39:56.430
It checks.
00:39:56.430 --> 00:40:02.070
And, similarly, I have to
show that if A v is zero,
00:40:02.070 --> 00:40:10.130
then A times any multiple,
say 12v is also zero.
00:40:10.130 --> 00:40:11.590
And how do I know that?
00:40:11.590 --> 00:40:14.900
Because I'm allowed to s-
bring that twelve outside.
00:40:14.900 --> 00:40:20.640
A number, a scaler can move
outside, so I have twelve A vs,
00:40:20.640 --> 00:40:21.740
twelve zeroes --
00:40:21.740 --> 00:40:23.960
I have zero.
00:40:23.960 --> 00:40:25.550
Okay.
00:40:25.550 --> 00:40:33.070
Just to -- it's really
critical to understand the --
00:40:33.070 --> 00:40:34.390
oh yeah.
00:40:34.390 --> 00:40:38.220
Here -- I was going to say,
understand what's the point
00:40:38.220 --> 00:40:39.470
of a vector space?
00:40:39.470 --> 00:40:44.790
Let me make that point by
changing the right-hand side.
00:40:44.790 --> 00:40:45.920
Oops.
00:40:45.920 --> 00:40:46.420
Okay.
00:40:46.420 --> 00:40:49.740
Let me change the right-hand
side to one two three four.
00:40:49.740 --> 00:40:51.632
Oh, okay.
00:40:51.632 --> 00:40:53.840
Why don't we do all of linear
algebra in one lecture,
00:40:53.840 --> 00:40:54.890
then we --
00:40:54.890 --> 00:40:55.900
okay.
00:40:55.900 --> 00:41:00.500
I would like to know the
solutions to this equation.
00:41:00.500 --> 00:41:01.670
For those four equations.
00:41:05.060 --> 00:41:06.660
So I have four equations.
00:41:06.660 --> 00:41:09.039
I have only three
unknowns, so if I
00:41:09.039 --> 00:41:10.830
don't have a pretty
special right-hand side
00:41:10.830 --> 00:41:12.640
there won't be any
solution at all.
00:41:12.640 --> 00:41:16.070
But that is a very
special right-hand side.
00:41:16.070 --> 00:41:21.450
And we know that there is
a solution, one zero zero.
00:41:21.450 --> 00:41:23.730
Were there any more solutions?
00:41:23.730 --> 00:41:27.320
And did they form
a vector space?
00:41:27.320 --> 00:41:28.600
Okay.
00:41:28.600 --> 00:41:31.550
So I'm asking two
questions there.
00:41:31.550 --> 00:41:35.530
One is, do -- so my right-hand
side now is not zero anymore.
00:41:35.530 --> 00:41:37.100
I'm not looking
at the null space
00:41:37.100 --> 00:41:40.070
because I changed from zeroes.
00:41:40.070 --> 00:41:45.930
So my first question is, do
the solutions, if there are any
00:41:45.930 --> 00:41:50.610
and there are, do
they form a subspace?
00:41:50.610 --> 00:41:53.270
Let's answer that
question first.
00:41:53.270 --> 00:41:54.210
Yes or no.
00:41:54.210 --> 00:41:59.640
Do I get a subspace if I
look at the solutions to --
00:41:59.640 --> 00:42:03.300
let me go back to x1 x2 x3.
00:42:03.300 --> 00:42:08.910
I'm looking at all the x-s, at
all those vectors in R^3 that
00:42:08.910 --> 00:42:10.380
solve A x -b.
00:42:10.380 --> 00:42:13.400
The only thing I've changed
is b isn't zero anymore.
00:42:17.360 --> 00:42:21.045
Do the x-s, the solutions,
form a vector space?
00:42:25.860 --> 00:42:32.670
The solutions to this
do not form a subspace.
00:42:32.670 --> 00:42:36.150
The solutions don't, because --
00:42:36.150 --> 00:42:38.460
how shall I see that?
00:42:38.460 --> 00:42:42.950
The zero vector is not a
solution, so I never even got
00:42:42.950 --> 00:42:43.700
started.
00:42:43.700 --> 00:42:46.090
The zero vector doesn't
solve this system.
00:42:46.090 --> 00:42:51.940
I can't -- solutions
can't be a vector space.
00:42:51.940 --> 00:42:55.480
Now what are they like?
00:42:55.480 --> 00:42:58.810
Well, we'll see this, but
let's do it for this example.
00:42:58.810 --> 00:43:02.200
So one zero zero was a solution.
00:43:02.200 --> 00:43:03.690
You saw that right away.
00:43:03.690 --> 00:43:05.310
Are there any other solutions?
00:43:05.310 --> 00:43:08.730
Can you tell me
a second solution
00:43:08.730 --> 00:43:10.430
to this system of equations?
00:43:10.430 --> 00:43:14.970
STUDENTS: 0 -1 1 STRANG: 0 -1 1.
00:43:14.970 --> 00:43:19.560
Boy, that's -- 0 -1 1.
00:43:19.560 --> 00:43:20.680
Yes.
00:43:20.680 --> 00:43:24.090
Because that says I take minus
this column plus this one
00:43:24.090 --> 00:43:24.790
and sure enough.
00:43:24.790 --> 00:43:27.730
That's right.
00:43:27.730 --> 00:43:30.750
So there are -- there's a
bunch of solutions here.
00:43:35.160 --> 00:43:37.300
But they're not a subspace.
00:43:37.300 --> 00:43:38.540
I'll tell you what it's like.
00:43:38.540 --> 00:43:41.460
It's like a plane that
doesn't go through the origin,
00:43:41.460 --> 00:43:43.930
or a line that doesn't
go through the origin.
00:43:43.930 --> 00:43:45.870
Maybe in this case
it's a line that
00:43:45.870 --> 00:43:47.870
doesn't go through
the origin, if I graft
00:43:47.870 --> 00:43:50.570
the solutions to A x equal B.
00:43:50.570 --> 00:43:53.550
So you -- I think
you've got the idea.
00:43:53.550 --> 00:43:56.780
Subspaces have to go
through the origin.
00:43:56.780 --> 00:44:01.590
If I'm looking at x-s, then
they'd better solve Ax=0.
00:44:01.590 --> 00:44:04.690
In a way I've got --
00:44:04.690 --> 00:44:10.800
my two subspaces that I --
talking about today are kind
00:44:10.800 --> 00:44:17.340
of the two ways I can tell
you what a -- about subspace.
00:44:17.340 --> 00:44:20.100
If I want to tell you
about the column space,
00:44:20.100 --> 00:44:24.780
I tell you a few columns and
I say take their combinations.
00:44:24.780 --> 00:44:27.330
Like I build up this subspace.
00:44:27.330 --> 00:44:31.440
I put in a few vectors, their
combinations make a subspace.
00:44:31.440 --> 00:44:35.010
Now, when I went to -- let me
come back to the one that is
00:44:35.010 --> 00:44:36.180
a subspace here.
00:44:40.980 --> 00:44:44.430
Here, when I talked
about the null space,
00:44:44.430 --> 00:44:46.960
I didn't tell you what's in it.
00:44:46.960 --> 00:44:49.350
We had to figure
out what was in it.
00:44:49.350 --> 00:44:53.290
What I told you was the
equations that I'm --
00:44:53.290 --> 00:44:55.180
that has to be satisfied.
00:44:55.180 --> 00:44:56.680
You see those --
00:44:56.680 --> 00:44:59.890
like, those are the two
natural ways to tell you
00:44:59.890 --> 00:45:02.460
what's in a subspace.
00:45:02.460 --> 00:45:06.730
I can either give you a few
vectors and say fill it out,
00:45:06.730 --> 00:45:08.660
take combinations --
00:45:08.660 --> 00:45:12.800
or I can give you a system of
equations, the requirements
00:45:12.800 --> 00:45:17.390
that the x-s have to satisfy.
00:45:17.390 --> 00:45:20.400
And both of those
ways produce subspaces
00:45:20.400 --> 00:45:25.080
and they're the important
ways to construct subspaces.
00:45:25.080 --> 00:45:29.730
Okay, so today's
lecture actually got,
00:45:29.730 --> 00:45:33.260
the essentials of
three point two,
00:45:33.260 --> 00:45:35.250
the idea of the null space.
00:45:35.250 --> 00:45:37.740
Now we have to
tackle, Wednesday,
00:45:37.740 --> 00:45:40.490
the job of how do
we actually get hold
00:45:40.490 --> 00:45:43.530
of that subspace in an
example that's bigger
00:45:43.530 --> 00:45:45.960
and we can't see it just by eye.
00:45:45.960 --> 00:45:48.770
Okay.
00:45:48.770 --> 00:45:57.190
See you Wednesday.
00:45:57.190 --> 00:45:58.740
Thanks.