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GILBERT STRANG: OK.

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We're coming to the point
where we need matrices.

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That's the point when we
have several equations,

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several differential
equations instead of just one.

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And it's a matrix that
does that coupling.

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So can I-- this won't be a
full course in linear algebra.

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That would be
available, you may know

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on, open courseware for 18.06.

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That's the linear
algebra course.

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But [INAUDIBLE] facts,
and why not just

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say them here in a few minutes?

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So I have a matrix.

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Well there's a matrix.

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That's a 3 by 3 matrix.

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And first I want to ask how
does it multiply a vector.

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So there it is multiplying
a vector, v1, v2, v3.

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And what's the result, key idea?

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It takes the answer
on the right-hand side

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is this number v1, times that
column, plus this number,

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that number times the second
column, plus the third number,

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the third number times
the third column,

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combination of the columns of a.

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That's what a times v is.

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That's what the notation of
matrix multiplication produces.

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That's really basic to see it
as a combination of columns.

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Now I want to build on that.

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That's one particular, if
you give me v1, v2, and v3,

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I know how to multiply it.

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I take the combination.

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Now I would like you to
think about the result

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from all v1, v2, and v3.

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If I take all those numbers, and
I get a whole lot of answers.

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They're all vectors,
the result of A times v

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is another vector,
Av, And I want

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to think about Av, those
outputs, for all inputs v.

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So I take v1, v2, v3 to
be [AUDIO OUT] numbers.

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And I get all combinations
of those three columns.

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And usually I would get the
whole 3-dimensional space.

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Usually I can
produce any vector,

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any output b1, b2, b3
from A times v. But not

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for this matrix,
not for this matrix.

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Because this matrix is,
you could say, deficient.

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That third column
there, 2, 3, 3,

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is obviously the sum of
columns one and column two.

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So this v3 times
that third column

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just produces something
that I could already get

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from column one and column two.

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That v3 times that column
three, I could x out.

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That's the same as column
one, plus column two

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for this matrix, not usually.

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And then so I only really have
a combination of two columns.

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It's a combination of three.

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But the third one was
dependent on the others.

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And it's really a
combination of two columns.

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So combinations of two
columns, two vectors

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in 3-dimensional
space produce a plane.

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I only get a plane.

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I don't get all of 3-dimensional
space, only a plane.

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And I call that plane
the column space,

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so the column space
of the matrix.

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So if you gave me
a different matrix,

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if you change this 3 to an
11, probably the column space

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now changes to--
for that matrix I

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think the column space would be
the whole 3-dimensional space.

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I get everything.

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But when this third column is
this the sum of the first two

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columns, it's not
giving me anything new.

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And the column space
is only a plane.

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And you can think of a matrix
where the column space is only

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a line, just one
independent column.

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OK.

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So that, we thought about this.

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[AUDIO OUT] is all
combinations of the columns.

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In other words, it's
all the results,

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all the outputs from A times
v. It's all the outputs

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from A times v. Those are the
combinations of the columns.

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So we can answer the most basic
question of linear algebra.

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When does Av equal b?

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Have [AUDIO OUT].

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When is there a v so
that I can solve this?

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When is there a v that
solves this equation?

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So it's a question about b.

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What is it about b that must
be true if this can be solved?

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Well this says that
equation is saying

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b is a combination
of the columns of a.

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So this has a
solution when b must

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be-- shall I say must
be in the column space.

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For that example,
only b's that where

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we can get a solution on
b's that are combinations

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of the first two columns.

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Because having the third
column at our disposal

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gives us no help.

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It doesn't give us anything new.

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[AUDIO OUT]

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It will be solvable
if b equalled 1, 1, 1.

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That's a combination of the
column, or if b equals 1, 2, 2.

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That's another simple
combination of the columns.

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Or if b equals 2, 3, 3.

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But I'm only, I'm
staying on a plane there.

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And most b's are off that plane.

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Now when there is a solution.

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All right.

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Now a second key idea
of linear algebra,

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can we do it in
this short video?

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I want to know about the
equation Av equals 0.

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So now I'm setting the
right-hand side to be 0.

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That's the 0 vector, 0, 0, 0.

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Does it have a solution?

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Does it have a solution?

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Let's take this example.

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1, 1, 1; 1, 2, 2; 2, 3, 3; now
I'm looking at the solutions

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when the right side is all 0.

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Does that have a solution?

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Is there a combination of those
three columns that gives 0?

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Well there is always
one combination.

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I could take 0, 0, and 0.

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I could take nothing,
0 of everything.

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0 of this column, 0 of that
column, 0 of the third column,

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would give me to
the 0 [AUDIO OUT].

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That solution is
always available.

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The big question is, is
there another solution.

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And here for this deficient,
singular, non-invertible

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matrix, there is.

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There is another solution.

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Let me just write it down.

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Let me put it in there.

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Do you see what the solution is?

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The third column is
the sum of those two.

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So if I want one of that
column, I should take minus 1

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in other column.

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So this is minus this
column, minus this column,

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plus this column
gives me the 0 column.

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That is a vector
in the null space.

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That's a solution to
Avn equals [AUDIO OUT].

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So the null space is all
solutions to Av equals 0.

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It's all the v's.

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The null space is
a bunch of v's.

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The column space
was a bunch of b's.

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It's just going to just
emphasize that difference.

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I was looking at
which b [AUDIO OUT].

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I wasn't paying attention to
what that solution was, just

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is there a solution.

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Then that b is in
the column space.

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I take b equals 0.

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I fixed that all important b.

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And now I'm looking
at the solutions.

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And here I find one.

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Can you find any more solutions?

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I think minus 10, minus 10, and
10 would be another solution.

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It's 10 times as much.

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And 0, 0, 0 is solution.

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[AUDIO OUT] line of solutions.

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We had a plane for
the column space.

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But we have a line
for the null space.

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Isn't that neat?

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One's a plane, one's
a line, dimension two

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plus dimension one.

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Two for the plane,
one for the line,

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adds to dimension three, the
dimension of the whole space.

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OK.

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That's a little going at in.

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All right.

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Now I ask, what
our all solutions?

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Complete solution
to Av equals, well

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let me choose some right-hand
side where there is a solution.

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Let me choose a
right-hand side, say

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if I add that column
and that column,

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I'll get Av-- maybe I'll
take two of that column

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plus one of that column.

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Two of the first column
with one of the second

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would be 3, 2 plus
that would be a 4,

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2 plus that would be another 4.

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OK.

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That's my b.

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It's a combination
of the columns.

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You saw me create it from
the first two columns.

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So now I ask, what
are all the solutions?

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It's in the column space.

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It's 2 times the first column,
plus the second column.

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But there may be
other solutions.

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So all solutions, a complete
solution, v complete

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is here's the key idea.

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And the point is that
it's the same that we know

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from differential equations.

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It's particular solution
plus any null solution.

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Plus all, you can
say all v null.

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Particular plus null solution.

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It's such an important concept
we just want to see it again.

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One particular
solution with that

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thing would be
particular, v particular

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could be-- 2-- how
did we produce that?

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Out of two these, plus one
of these, plus zero of that.

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So v particular
could be 2, 1, 0.

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It works for that particular
b, two of the first column,

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one of the second.

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Now then we could add in
anything in the null solution.

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So we have infinitely
many solutions here.

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We've got one solution
plus added to that,

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a whole line of solutions.

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This, all the null space,
would be all vectors like that.

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OK.

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That's the picture
that we've seen

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for differential equations.

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And I just want to bring it
out again for matrix equations,

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using the language
of linear algebra.

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That's what I'm
introducing here.

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I have one particular
solution, plus anything

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in the null [AUDIO OUT]
space of vectors that

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is the heart of linear algebra.

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Thank you.