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Hi.

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This is the first lecture in
MIT's course 18.06,

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linear algebra,
and I'm Gilbert Strang.

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The text for the course is this
book, Introduction to Linear

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Algebra.

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And the course web page,
which has got a lot of

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exercises from the past,
MatLab codes,

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the syllabus for the course,
is web.mit.edu/18.06.

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And this is the first lecture,
lecture one.

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So, and later we'll give the
web address for viewing these,

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videotapes.

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Okay, so what's in the first
lecture?

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This is my plan.

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The fundamental problem of
linear algebra,

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which is to solve a system of
linear equations.

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So let's start with a case when
we have some number of

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equations, say n equations and n
unknowns.

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So an equal number of equations
and unknowns.

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That's the normal,
nice case.

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And what I want to do is --
with examples,

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of course -- to describe,
first, what I call the Row

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picture.

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That's the picture of one
equation at a time.

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It's the picture you've seen
before in two by two equations

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where lines meet.

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So in a minute,
you'll see lines meeting.

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The second picture,
I'll put a star beside that,

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because that's such an
important one.

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And maybe new to you is the
picture -- a column at a time.

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And those are the rows and
columns of a matrix.

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So the third -- the algebra way
to look at the problem is the

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matrix form and using a matrix
that I'll call A.

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Okay, so can I do an example?

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The whole semester will be
examples and then see what's

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going on with the example.

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So, take an example.

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Two equations,
two unknowns.

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So let me take 2x -y =0,
let's say.

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And -x 2y=3.

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Okay.
let me -- I can even say right

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away -- what's the matrix,
that is, what's the coefficient

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matrix?

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The matrix that involves these
numbers --

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a matrix is just a rectangular
array of numbers.

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Here it's two rows and two
columns, so 2 and -- minus 1 in

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the first row minus 1 and 2 in
the second row,

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that's the matrix.

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And the right-hand -- the,
unknown -- well,

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we've got two unknowns.

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So we've got a vector,
with two components,

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x and x, and we've got two
right-hand sides that go into a

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vector 0 3.

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I couldn't resist writing the
matrix form, right -- even

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before the pictures.

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So I always will think of this
as the matrix A,

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the matrix of coefficients,
then there's a vector of

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unknowns.

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Here we've only got two
unknowns.

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Later we'll have any number of
unknowns.

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And that vector of unknowns,
well I'll often -- I'll make

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that x -- extra bold.

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A and the right-hand side is
also a vector that I'll always

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call b.

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So linear equations are A x
equal b and the idea now is to

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solve this particular example
and then step back to see the

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bigger picture.

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Okay, what's the picture for
this example,

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the Row picture?

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Okay, so here comes the Row
picture.

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So that means I take one row at
a time and I'm drawing here the

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xy plane and I'm going to plot
all the points that satisfy that

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first equation.

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So I'm looking at all the
points that satisfy 2x-y =0.

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It's often good to start with
which point on the horizontal

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line --
on this horizontal line,

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y is zero.

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The x axis has y as zero and
that -- in this case,

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actually, then x is zero.

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So the point,
the origin -- the point with

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coordinates (0,0) is on the
line.

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It solves that equation.

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Okay, tell me in --
well, I guess I have to tell

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you another point that solves
this same equation.

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Let me suppose x is one,
so I'll take x to be one.

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Then y should be two,
right?

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So there's the point one two
that also solves this equation.

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And I could put in more points.

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But, but let me put in all the
points at once,

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because they all lie on a
straight line.

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This is a linear equation and
that word linear got the letters

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for line in it.

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That's the equation -- this is
the line that ...

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of solutions to 2x-y=0 my first
row, first equation.

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So typically,
maybe, x equal a half,

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y equal one will work.

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And sure enough it does.

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Okay, that's the first one.

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Now the second one is not going
to go through the origin.

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It's always important.

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Do we go through the origin or
not?

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In this case,
yes, because there's a zero

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over there.

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In this case we don't go
through the origin,

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because if x and y are zero,
we don't get three.

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So, let me again say suppose y
is zero, what x do we actually

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get?

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If y is zero,
then I get x is minus three.

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So if y is zero,
I go along minus three.

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So there's one point on this
second line.

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Now let me say,
well, suppose x is minus one --

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just to take another x.

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If x is minus one,
then this is a one and I think

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y should be a one,
because if x is minus one,

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then I think y should be a one
and we'll get that point.

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Is that right?

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If x is minus one,
that's a one.

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If y is a one,
that's a two and the one and

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the two make three and that
point's on the equation.

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Okay.

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Now, I should just draw the
line, right, connecting those

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two points at -- that will give
me the whole line.

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And if I've done this
reasonably well,

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I think it's going to happen to
go through -- well,

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not happen -- it was arranged
to go through that point.

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So I think that the second line
is this one, and this is the

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all-important point that lies on
both lines.

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Shall we just check that that
point which is the point x equal

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one and y was two,
right?

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That's the point there and
that, I believe,

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solves both equations.

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Let's just check this.

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If x is one,
I have a minus one plus four

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equals three,
okay.

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Apologies for drawing this
picture that you've seen before.

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But this -- seeing the row
picture --

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first of all,
for n equal 2,

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two equations and two unknowns,
it's the right place to start.

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Okay.

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So we've got the solution.

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The point that lies on both
lines.

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Now can I come to the column
picture?

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Pay attention,
this is the key point.

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So the column picture.

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I'm now going to look at the
columns of the matrix.

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I'm going to look at this part
and this part.

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I'm going to say that the x
part is really x times -- you

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see, I'm putting the two -- I'm
kind of getting the two

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equations at once --
that part and then I have a y

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and in the first equation it's
multiplying a minus one and in

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the second equation a two,
and on the right-hand side,

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zero and three.

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You see, the columns of the
matrix, the columns of A are

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here and the right-hand side b
is there.

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And now what is the equation
asking for?

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It's asking us to find --
somehow to combine that vector

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and this one in the right
amounts to get that one.

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It's asking us to find the
right linear combination -- this

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is called a linear combination.

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And it's the most fundamental
operation in the whole course.

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It's a linear combination of
the columns.

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That's what we're seeing on the
left side.

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Again, I don't want to write
down a big definition.

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You can see what it is.

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There's column one,
there's column two.

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I multiply by some numbers and
I add.

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That's a combination -- a
linear combination and I want to

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make those numbers the right
numbers to produce zero three.

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Okay.

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Now I want to draw a picture
that, represents what this --

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this is algebra.

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What's the geometry,
what's the picture that goes

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with it?

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Okay.

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So again, these vectors have
two components,

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so I better draw a picture like
that.

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So can I put down these
columns?

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I'll draw these columns as they
are, and then I'll do a

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combination of them.

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So the first column is over two
and down one,

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right?

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So there's the first column.

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The first column.

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Column one.

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It's the vector two minus one.

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The second column is -- minus
one is the first component and

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up two.

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It's here.

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There's column two.

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So this, again,
you see what its components

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are.

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Its components are minus one,
two.

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Good.

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That's this guy.

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Now I have to take a
combination.

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What combination shall I take?

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Why not the right combination,
what the hell?

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Okay.

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So the combination I'm going to
take is the right one to produce

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zero three and then we'll see it
happen in the picture.

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So the right combination is to
take x as one of those and two

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of these.

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It's because we already know
that that's the right x and y,

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so why not take the correct
combination here and see it

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happen?

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Okay, so how do I picture this
linear combination?

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So I start with this vector
that's already here -- so that's

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one of column one,
that's one times column one,

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right there.

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And now I want to add on --
so I'm going to hook the next

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vector onto the front of the
arrow will start the next vector

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and it will go this way.

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So let's see,
can I do it right?

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If I added on one of these
vectors, it would go left one

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and up two, so we'd go left one
and up two, so it would probably

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get us to there.

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Maybe I'll do dotted line for
that.

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Okay?

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That's one of column two tucked
onto the end,

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but I wanted to tuck on two of
column two.

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So that -- the second one --
we'll go up left one and up two

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also.

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It'll probably end there.

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And there's another one.

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So what I've put in here is two
of column two.

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Added on.

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And where did I end up?

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What are the coordinates of
this result?

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What do I get when I take one
of this plus two of that?

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I do get that,
of course.

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There it is,
x is zero, y is three,

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that's b.

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That's the answer we wanted.

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And how do I do it?

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You see I do it just like the
first component.

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I have a two and a minus two
that produces a zero,

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and in the second component I
have a minus one and a four,

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they combine to give the three.

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But look at this picture.

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So here's our key picture.

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I combine this column and this
column to get this guy.

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That was the b.

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That's the zero three.

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Okay.

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So that idea of linear
combination is crucial,

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and also -- do we want to think
about this question?

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Sure, why not.

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What are all the combinations?

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If I took -- can I go back to
xs and ys?

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This is a question for really
-- it's going to come up over

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and over, but why don't we see
it once now?

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If I took all the xs and all
the ys, all the combinations,

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what would be all the results?

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And, actually,
the result would be that I

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could get any right-hand side at
all.

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The combinations of this and
this would fill the whole plane.

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You can tuck that away.

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We'll, explore it further.

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But this idea of what linear
combination gives b and what do

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all the linear combinations
give,

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what are all the possible,
achievable right-hand sides be

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-- that's going to be basic.

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Okay.

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Can I move to three equations
and three unknowns?

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Because it's easy to picture
the two by two case.

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Let me do a three by three
example.

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Okay, I'll sort of start it the
same way,

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say maybe 2x-y and maybe I'll
take no zs as a zero and maybe a

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-x 2y and maybe a -z as a -- oh,
let me make that a minus one

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and, just for variety let me
take, -3z, -3ys,

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I should keep the ys in that
line, and 4zs is,

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say, 4.

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Okay.

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That's three equations.

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I'm in three dimensions,
x, y, z.

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And, I don't have a solution
yet.

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So I want to understand the
equations and then solve them.

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Okay.

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So how do I you understand
them?

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The row picture one way.

280
00:16:32 --> 00:16:38
The column picture is another
very important way.

281
00:16:38 --> 00:16:43
Just let's remember the matrix
form, here, because that's easy.

282
00:16:43 --> 00:16:46
The matrix form -- what's our
matrix A?

283
00:16:46 --> 00:16:51
Our matrix A is this right-hand
side, the two and the minus one

284
00:16:51 --> 00:16:56
and the zero from the first row,
the minus one and the two and

285
00:16:56 --> 00:16:59
the minus one from the second
row,

286
00:16:59 --> 00:17:04
the zero, the minus three and
the four from the third row.

287
00:17:04 --> 00:17:07
So it's a three by three
matrix.

288
00:17:07 --> 00:17:10
Three equations,
three unknowns.

289
00:17:10 --> 00:17:12
And what's our right-hand side?

290
00:17:12 --> 00:17:16
Of course, it's the vector,
zero minus one,

291
00:17:16 --> 00:17:16.63
four.

292
00:17:16.63 --> 00:17:17
Okay.

293
00:17:17 --> 00:17:21.13
So that's the way,
well, that's the short-hand to

294
00:17:21.13 --> 00:17:24
write out the three equations.

295
00:17:24 --> 00:17:30
But it's the picture that I'm
looking for today.

296
00:17:30 --> 00:17:33.96
Okay, so the row picture.

297
00:17:33.96 --> 00:17:39
All right, so I'm in three
dimensions, x,

298
00:17:39 --> 00:17:40
y and z.

299
00:17:40 --> 00:17:46.97
And I want to take those
equations one at a time and ask

300
00:17:46.97 --> 00:17:55
-- and make a picture of all the
points that satisfy --

301
00:17:55 --> 00:17:57
let's take equation number two.

302
00:17:57 --> 00:18:02
If I make a picture of all the
points that satisfy -- all the

303
00:18:02 --> 00:18:05
x, y, z points that solve this
equation -- well,

304
00:18:05 --> 00:18:08
first of all,
the origin is not one of them.

305
00:18:08 --> 00:18:12
x, y, z -- it being 0,
0, 0 would not solve that

306
00:18:12 --> 00:18:14
equation.

307
00:18:14 --> 00:18:18
So what are some points that do
solve the equation?

308
00:18:18 --> 00:18:22
Let's see, maybe if x is one,
y and z could be zero.

309
00:18:22 --> 00:18:24
That would work,
right?

310
00:18:24 --> 00:18:26
So there's one point.

311
00:18:26 --> 00:18:30
I'm looking at this second
equation, here,

312
00:18:30 --> 00:18:32
just, to start with.

313
00:18:32 --> 00:18:34
Let's see.

314
00:18:34 --> 00:18:38
Also, I guess,
if z could be one,

315
00:18:38 --> 00:18:44
x and y could be zero,
so that would just go straight

316
00:18:44 --> 00:18:46.38
up that axis.

317
00:18:46.38 --> 00:18:51
And, probably I'd want a third
point here.

318
00:18:51 --> 00:18:56
Let me take x to be zero,
z to be zero,

319
00:18:56 --> 00:19:02
then y would be minus a half,
right?

320
00:19:02 --> 00:19:07
So there's a third point,
somewhere -- oh my -- okay.

321
00:19:07 --> 00:19:07
Let's see.

322
00:19:07 --> 00:19:13
I want to put in all the points
that satisfy that equation.

323
00:19:13 --> 00:19:17
Do you know what that bunch of
points will be?

324
00:19:17 --> 00:19:18
It's a plane.

325
00:19:18 --> 00:19:22
If we have a linear equation,
then, fortunately,

326
00:19:22 --> 00:19:28
the graph of the thing,
the plot of all the points that

327
00:19:28 --> 00:19:30
solve it are a plane.

328
00:19:30 --> 00:19:36
These three points determine a
plane, but your lecturer is not

329
00:19:36 --> 00:19:41
Rembrandt and the art is going
to be the weak point here.

330
00:19:41 --> 00:19:45
So I'm just going to draw a
plane, right?

331
00:19:45 --> 00:19:48
There's a plane somewhere.

332
00:19:48 --> 00:19:49
That's my plane.

333
00:19:49 --> 00:19:54
That plane is all the points
that solves this guy.

334
00:19:54 --> 00:19:56
Then, what about this one?

335
00:19:56 --> 00:19:59
Two x minus y plus zero z.

336
00:19:59 --> 00:20:01
So z actually can be anything.

337
00:20:01 --> 00:20:06
Again, it's going to be another
plane.

338
00:20:06 --> 00:20:11
Each row in a three by three
problem gives us a plane in

339
00:20:11 --> 00:20:12.75
three dimensions.

340
00:20:12.75 --> 00:20:18
So this one is going to be some
other plane -- maybe I'll try to

341
00:20:18 --> 00:20:20
draw it like this.

342
00:20:20 --> 00:20:23
And those two planes meet in a
line.

343
00:20:23 --> 00:20:30
So if I have two equations,
just the first two equations in

344
00:20:30 --> 00:20:34
three dimensions,
those give me a line.

345
00:20:34 --> 00:20:38.73
The line where those two planes
meet.

346
00:20:38.73 --> 00:20:43
And now, the third guy is a
third plane.

347
00:20:43 --> 00:20:45
And it goes somewhere.

348
00:20:45 --> 00:20:51
Okay, those three things meet
in a point.

349
00:20:51 --> 00:20:55
Now I don't know where that
point is, frankly.

350
00:20:55 --> 00:20:57
But -- linear algebra will find
it.

351
00:20:57 --> 00:21:00
The main point is that the
three planes,

352
00:21:00 --> 00:21:04
because they're not parallel,
they're not special.

353
00:21:04 --> 00:21:09.45
They do meet in one point and
that's the solution.

354
00:21:09.45 --> 00:21:14
But, maybe you can see that
this row picture is getting a

355
00:21:14 --> 00:21:16
little hard to see.

356
00:21:16 --> 00:21:21
The row picture was a cinch
when we looked at two lines

357
00:21:21 --> 00:21:21
meeting.

358
00:21:21 --> 00:21:26
When we look at three planes
meeting, it's not so clear and

359
00:21:26 --> 00:21:32
in four dimensions probably a
little less clear.

360
00:21:32 --> 00:21:35.2
So, can I quit on the row
picture?

361
00:21:35.2 --> 00:21:40
Or quit on the row picture
before I've successfully found

362
00:21:40 --> 00:21:43
the point where the three planes
meet?

363
00:21:43 --> 00:21:49
All I really want to see is
that the row picture consists of

364
00:21:49 --> 00:21:54
three planes and,
if everything works right,

365
00:21:54 --> 00:21:59
three planes meet in one point
and that's a solution.

366
00:21:59 --> 00:22:03
Now, you can tell I prefer the
column picture.

367
00:22:03 --> 00:22:06
Okay, so let me take the column
picture.

368
00:22:06 --> 00:22:12.07
That's x times -- so there were
two xs in the first equation

369
00:22:12.07 --> 00:22:16
minus one x is,
and no xs in the third.

370
00:22:16 --> 00:22:19
It's just the first column of
that.

371
00:22:19 --> 00:22:21
And how many ys are there?

372
00:22:21 --> 00:22:27
There's minus one in the first
equations, two in the second and

373
00:22:27 --> 00:22:30
maybe minus three in the third.

374
00:22:30 --> 00:22:33
Just the second column of my
matrix.

375
00:22:33 --> 00:22:37
And z times no zs minus one zs
and four zs.

376
00:22:37 --> 00:22:43
And it's those three columns,
right, that I have to combine

377
00:22:43 --> 00:22:48
to produce the right-hand side,
which is zero minus one four.

378
00:22:48 --> 00:22:49
Okay.

379
00:22:49 --> 00:22:52.87
So what have we got on this
left-hand side?

380
00:22:52.87 --> 00:22:54
A linear combination.

381
00:22:54 --> 00:22:58
It's a linear combination now
of three vectors,

382
00:22:58 --> 00:23:03
and they happen to be -- each
one is a three dimensional

383
00:23:03 --> 00:23:07
vector,
so we want to know what

384
00:23:07 --> 00:23:10
combination of those three
vectors produces that one.

385
00:23:10 --> 00:23:13.64
Shall I try to draw the column
picture, then?

386
00:23:13.64 --> 00:23:17
So, since these vectors have
three components -- so it's some

387
00:23:17 --> 00:23:22
multiple -- let me draw in the
first column as before --

388
00:23:22 --> 00:23:25
x is two and y is minus one.

389
00:23:25 --> 00:23:28
Maybe there is the first
column.

390
00:23:28 --> 00:23:35
y -- the second column has
maybe a minus one and a two and

391
00:23:35 --> 00:23:41
the y is a minus three,
somewhere, there possibly,

392
00:23:41 --> 00:23:42
column two.

393
00:23:42 --> 00:23:49
And the third column has --
no zero minus one four,

394
00:23:49 --> 00:23:51
so how shall I draw that?

395
00:23:51 --> 00:23:54
So this was the first
component.

396
00:23:54 --> 00:23:58.73
The second component was a
minus one.

397
00:23:58.73 --> 00:24:00
Maybe up here.

398
00:24:00 --> 00:24:05
That's column three,
that's the column zero minus

399
00:24:05 --> 00:24:07
one and four.

400
00:24:07 --> 00:24:08
This guy.

401
00:24:08 --> 00:24:11
So, again, what's my problem?

402
00:24:11 --> 00:24:18
What this equation is asking me
to do is to combine these three

403
00:24:18 --> 00:24:24
vectors with a right combination
to produce this one.

404
00:24:24 --> 00:24:29
Well, you can see what the
right combination is,

405
00:24:29 --> 00:24:33
because in this special
problem,

406
00:24:33 --> 00:24:38
specially chosen by the
lecturer, that right-hand side

407
00:24:38 --> 00:24:43.77
that I'm trying to get is
actually one of these columns.

408
00:24:43.77 --> 00:24:46
So I know how to get that one.

409
00:24:46 --> 00:24:48
So what's the solution?

410
00:24:48 --> 00:24:51
What combination will work?

411
00:24:51 --> 00:24:55
I just want one of these and
none of these.

412
00:24:55 --> 00:25:00
So x should be zero,
y should be zero and z should

413
00:25:00 --> 00:25:00
be one.

414
00:25:00 --> 00:25:02
That's the combination.

415
00:25:02 --> 00:25:06
One of those is obviously the
right one.

416
00:25:06 --> 00:25:10.95
Column three is actually the
same as b in this particular

417
00:25:10.95 --> 00:25:11.65
problem.

418
00:25:11.65 --> 00:25:17
I made it work that way just so
we would get an answer,

419
00:25:17 --> 00:25:23
(0,0,1), so somehow that's the
point where those three planes

420
00:25:23 --> 00:25:26
met and I couldn't see it
before.

421
00:25:26 --> 00:25:32
Of course, I won't always be
able to see it from the column

422
00:25:32 --> 00:25:34
picture, either.

423
00:25:34 --> 00:25:38
It's the next lecture,
actually,

424
00:25:38 --> 00:25:43
which is about elimination,
which is the systematic way

425
00:25:43 --> 00:25:49
that everybody -- every bit of
software, too -- production,

426
00:25:49 --> 00:25:53
large-scale software would
solve the equations.

427
00:25:53 --> 00:25:56
So the lecture that's coming
up.

428
00:25:56 --> 00:26:02
If I was to add that to the
syllabus, will be about how to

429
00:26:02 --> 00:26:05
find x, y, z in all cases.

430
00:26:05 --> 00:26:11
Can I just think again,
though, about the big picture?

431
00:26:11 --> 00:26:17
By the big picture I mean let's
keep this same matrix on the

432
00:26:17 --> 00:26:24.12
left but imagine that we have a
different right-hand side.

433
00:26:24.12 --> 00:26:29
Oh, let me take a different
right-hand side.

434
00:26:29 --> 00:26:33
So I'll change that right-hand
side to something that actually

435
00:26:33 --> 00:26:35
is also pretty special.

436
00:26:35 --> 00:26:38
Let me change it to -- if I add
those first two columns,

437
00:26:38 --> 00:26:42
that would give me a one and a
one and a minus three.

438
00:26:42 --> 00:26:44
There's a very special
right-hand side.

439
00:26:44 --> 00:26:48.88
I just cooked it up by adding
this one to this one.

440
00:26:48.88 --> 00:26:54.17
Now, what's the solution with
this new right-hand side?

441
00:26:54.17 --> 00:26:59
The solution with this new
right-hand side is clear.

442
00:26:59 --> 00:27:02
took one of these and none of
those.

443
00:27:02 --> 00:27:06
So actually,
it just changed around to this

444
00:27:06 --> 00:27:11
when I took this new right-hand
side.

445
00:27:11 --> 00:27:11
Okay.

446
00:27:11 --> 00:27:17
So in the row picture,
I have three different planes,

447
00:27:17 --> 00:27:21
three new planes meeting now at
this point.

448
00:27:21 --> 00:27:27
In the column picture,
I have the same three columns,

449
00:27:27 --> 00:27:32
but now I'm combining them to
produce this guy,

450
00:27:32 --> 00:27:38
and it turned out that column
one plus column two which would

451
00:27:38 --> 00:27:43
be somewhere -- there is the
right column -- one of this and

452
00:27:43 --> 00:27:46
one of this would give me the
new b.

453
00:27:46 --> 00:27:46
Okay.

454
00:27:46 --> 00:27:49
So we squeezed in an extra
example.

455
00:27:49 --> 00:27:54.75
But now think about all bs,
all right-hand sides.

456
00:27:54.75 --> 00:28:02.06
Can I solve these equations for
every right-hand side?

457
00:28:02.06 --> 00:28:05.37
Can I ask that question?

458
00:28:05.37 --> 00:28:09
So that's the algebra question.

459
00:28:09 --> 00:28:13
Can I solve A x=b for every b?

460
00:28:13 --> 00:28:16
Let me write that down.

461
00:28:16 --> 00:28:24
Can I solve A x =b for every
right-hand side b?

462
00:28:24 --> 00:28:26
I mean, is there a solution?

463
00:28:26 --> 00:28:30
And then, if there is,
elimination will give me a way

464
00:28:30 --> 00:28:31
to find it.

465
00:28:31 --> 00:28:35
I really wanted to ask,
is there a solution for every

466
00:28:35 --> 00:28:37
right-hand side?

467
00:28:37 --> 00:28:41
So now, can I put that in
different words -- in this

468
00:28:41 --> 00:28:43
linear combination words?

469
00:28:43 --> 00:28:54
So in linear combination words,
do the linear combinations of

470
00:28:54 --> 00:29:00
the columns fill three
dimensional space?

471
00:29:00 --> 00:29:09
Every b means all the bs in
three dimensional space.

472
00:29:09 --> 00:29:20
Do you see that I'm just asking
the same question in different

473
00:29:20 --> 00:29:22
words?

474
00:29:22 --> 00:29:27
Solving A x -- A x -- that's
very important.

475
00:29:27 --> 00:29:32
A times x -- when I multiply a
matrix by a vector,

476
00:29:32 --> 00:29:36
I get a combination of the
columns.

477
00:29:36 --> 00:29:39
I'll write that down in a
moment.

478
00:29:39 --> 00:29:46
But in my column picture,
that's really what I'm doing.

479
00:29:46 --> 00:29:55.29
I'm taking linear combinations
of these three columns and I'm

480
00:29:55.29 --> 00:29:57
trying to find b.

481
00:29:57 --> 00:30:04
And, actually,
the answer for this matrix will

482
00:30:04 --> 00:30:05
be yes.

483
00:30:05 --> 00:30:14
For this matrix A -- for these
columns, the answer is yes.

484
00:30:14 --> 00:30:19
This matrix -- that I chose for
an example is a good matrix.

485
00:30:19 --> 00:30:21
A non-singular matrix.

486
00:30:21 --> 00:30:23
An invertible matrix.

487
00:30:23 --> 00:30:27
Those will be the matrices that
we like best.

488
00:30:27 --> 00:30:33
There could be other -- and we
will see other matrices where

489
00:30:33 --> 00:30:37
the answer becomes,
no -- oh, actually,

490
00:30:37 --> 00:30:40
you can see when it would
become no.

491
00:30:40 --> 00:30:42
What could go wrong?

492
00:30:42 --> 00:30:48
How could it go wrong that out
of these -- out of three columns

493
00:30:48 --> 00:30:53
and all their combinations --
when would I not be able to

494
00:30:53 --> 00:30:55
produce some b off here?

495
00:30:55 --> 00:30:58
When could it go wrong?

496
00:30:58 --> 00:31:02
Do you see that the
combinations -- let me say when

497
00:31:02 --> 00:31:05
it goes wrong.

498
00:31:05 --> 00:31:10
If these three columns all lie
in the same plane,

499
00:31:10 --> 00:31:16
then their combinations will
lie in that same plane.

500
00:31:16 --> 00:31:18.76
So then we're in trouble.

501
00:31:18.76 --> 00:31:25
If the three columns of my
matrix -- if those three vectors

502
00:31:25 --> 00:31:31
happen to lie in the same plane
-- for example,

503
00:31:31 --> 00:31:36
if column three is just the sum
of column one and column two,

504
00:31:36 --> 00:31:38
I would be in trouble.

505
00:31:38 --> 00:31:42
That would be a matrix A where
the answer would be no,

506
00:31:42 --> 00:31:47
because the combinations -- if
column three is in the same

507
00:31:47 --> 00:31:52
plane as column one and two,
I don't get anything new from

508
00:31:52 --> 00:31:53
that.

509
00:31:53 --> 00:31:58
All the combinations are in the
plane and only right-hand sides

510
00:31:58 --> 00:32:01
b that I could get would be the
ones in that plane.

511
00:32:01 --> 00:32:05
So I could solve it for some
right-hand sides,

512
00:32:05 --> 00:32:09
when b is in the plane,
but most right-hand sides would

513
00:32:09 --> 00:32:11
be out of the plane and
unreachable.

514
00:32:11 --> 00:32:14
So that would be a singular
case.

515
00:32:14 --> 00:32:17
The matrix would be not
invertible.

516
00:32:17 --> 00:32:22.07
There would not be a solution
for every b.

517
00:32:22.07 --> 00:32:25
The answer would become no for
that.

518
00:32:25 --> 00:32:26
Okay.

519
00:32:26 --> 00:32:32
I don't know -- shall we take
just a little shot at thinking

520
00:32:32 --> 00:32:34
about nine dimensions?

521
00:32:34 --> 00:32:39
Imagine that we have vectors
with nine components.

522
00:32:39 --> 00:32:45
Well, it's going to be hard to
visualize those.

523
00:32:45 --> 00:32:48
I don't pretend to do it.

524
00:32:48 --> 00:32:50
But somehow,
pretend you do.

525
00:32:50 --> 00:32:55
Pretend we have -- if this was
nine equations and nine

526
00:32:55 --> 00:32:59
unknowns, then we would have
nine columns,

527
00:32:59 --> 00:33:04
and each one would be a vector
in nine-dimensional space and we

528
00:33:04 --> 00:33:09
would be looking at their linear
combinations.

529
00:33:09 --> 00:33:13
So we would be having the
linear combinations of nine

530
00:33:13 --> 00:33:16
vectors in nine-dimensional
space, and we would be trying to

531
00:33:16 --> 00:33:20
find the combination that hit
the correct right-hand side b.

532
00:33:20 --> 00:33:23
And we might also ask the
question can we always do it?

533
00:33:23 --> 00:33:26
Can we get every right-hand
side b?

534
00:33:26 --> 00:33:30
And certainly it will depend on
those nine columns.

535
00:33:30 --> 00:33:34
Sometimes the answer will be
yes -- if I picked a random

536
00:33:34 --> 00:33:36
matrix, it would be yes,
actually.

537
00:33:36 --> 00:33:40
If I used MatLab and just used
the random command,

538
00:33:40 --> 00:33:44
picked out a nine by nine
matrix, I guarantee it would be

539
00:33:44 --> 00:33:44
good.

540
00:33:44 --> 00:33:48
It would be non-singular,
it would be invertible,

541
00:33:48 --> 00:33:50.28
all beautiful.

542
00:33:50.28 --> 00:33:57
But if I choose those columns
so that they're not independent,

543
00:33:57 --> 00:34:04.97
so that the ninth column is the
same as the eighth column,

544
00:34:04.97 --> 00:34:12
then it contributes nothing new
and there would be right-hand

545
00:34:12 --> 00:34:16
sides b that I couldn't get.

546
00:34:16 --> 00:34:21
Can you sort of think about
nine vectors in nine-dimensional

547
00:34:21 --> 00:34:23
space an take their
combinations?

548
00:34:23 --> 00:34:28
That's really the central
thought -- that you get kind of

549
00:34:28 --> 00:34:30
used to in linear algebra.

550
00:34:30 --> 00:34:33.27
Even though you can't really
visualize it,

551
00:34:33.27 --> 00:34:37
you sort of think you can after
a while.

552
00:34:37 --> 00:34:42
Those nine columns and all
their combinations may very well

553
00:34:42 --> 00:34:45
fill out the whole
nine-dimensional space.

554
00:34:45 --> 00:34:50
But if the ninth column
happened to be the same as the

555
00:34:50 --> 00:34:56
eighth column and gave nothing
new, then probably what it would

556
00:34:56 --> 00:35:01
fill out would be -- I hesitate
even to say this --

557
00:35:01 --> 00:35:06
it would be a sort of a plane
-- an eight dimensional plane

558
00:35:06 --> 00:35:08
inside nine-dimensional space.

559
00:35:08 --> 00:35:12
And it's those eight
dimensional planes inside

560
00:35:12 --> 00:35:17
nine-dimensional space that we
have to work with eventually.

561
00:35:17 --> 00:35:22
For now, let's stay with a nice
case where the matrices work,

562
00:35:22 --> 00:35:27
we can get every right-hand
side b and here we see how to do

563
00:35:27 --> 00:35:29
it with columns.

564
00:35:29 --> 00:35:30.04
Okay.

565
00:35:30.04 --> 00:35:37
There was one step which I
realized I was saying in words

566
00:35:37 --> 00:35:41
that I now want to write in
letters.

567
00:35:41 --> 00:35:48
Because I'm coming back to the
matrix form of the equation,

568
00:35:48 --> 00:35:52
so let me write it here.

569
00:35:52 --> 00:35:58
The matrix form of my equation,
of my system is some matrix A

570
00:35:58 --> 00:36:02
times some vector x equals some
right-hand side b.

571
00:36:02 --> 00:36:03
Okay.

572
00:36:03 --> 00:36:05
So this is a multiplication.

573
00:36:05 --> 00:36:06
A times x.

574
00:36:06 --> 00:36:10
Matrix times vector,
and I just want to say how do

575
00:36:10 --> 00:36:13
you multiply a matrix by a
vector?

576
00:36:13 --> 00:36:18
Okay, so I'm just going to
create a matrix --

577
00:36:18 --> 00:36:25
let me take two five one three
-- and let me take a vector x to

578
00:36:25 --> 00:36:26
be, say, 1and 2.

579
00:36:26 --> 00:36:30
How do I multiply a matrix by a
vector?

580
00:36:30 --> 00:36:36
But just think a little bit
about matrix notation and how to

581
00:36:36 --> 00:36:39
do that in multiplication.

582
00:36:39 --> 00:36:45
So let me say how I multiply a
matrix by a vector.

583
00:36:45 --> 00:36:48
Actually, there are two ways to
do it.

584
00:36:48 --> 00:36:51
Let me tell you my favorite
way.

585
00:36:51 --> 00:36:53
It's columns again.

586
00:36:53 --> 00:36:55
It's a column at a time.

587
00:36:55 --> 00:36:59
For me, this matrix
multiplication says I take one

588
00:36:59 --> 00:37:04
of that column and two of that
column and add.

589
00:37:04 --> 00:37:09
So this is the way I would
think of it is one of the first

590
00:37:09 --> 00:37:13
column and two of the second
column and let's just see what

591
00:37:13 --> 00:37:14
we get.

592
00:37:14 --> 00:37:18
So in the first component I'm
getting a two and a ten.

593
00:37:18 --> 00:37:20
I'm getting a twelve there.

594
00:37:20 --> 00:37:24
In the second component I'm
getting a one and a six,

595
00:37:24 --> 00:37:26
I'm getting a seven.

596
00:37:26 --> 00:37:31
So that matrix times that
vector is twelve seven.

597
00:37:31 --> 00:37:35
Now, you could do that another
way.

598
00:37:35 --> 00:37:38
You could do it a row at a
time.

599
00:37:38 --> 00:37:44
And you would get this twelve
-- and actually I pretty much

600
00:37:44 --> 00:37:47
did it here -- this way.

601
00:37:47 --> 00:37:51
Two -- I could take that row
times my vector.

602
00:37:51 --> 00:37:53
This is the idea of a dot
product.

603
00:37:53 --> 00:37:57.91
This vector times this vector,
two times one plus five times

604
00:37:57.91 --> 00:37:59
two is the twelve.

605
00:37:59 --> 00:38:03
This vector times this vector
-- one times one plus three

606
00:38:03 --> 00:38:05
times two is the seven.

607
00:38:05 --> 00:38:11
So I can do it by rows,
and in each row times my x is

608
00:38:11 --> 00:38:14
what I'll later call a dot
product.

609
00:38:14 --> 00:38:18
But I also like to see it by
columns.

610
00:38:18 --> 00:38:23
I see this as a linear
combination of a column.

611
00:38:23 --> 00:38:25
So here's my point.

612
00:38:25 --> 00:38:31
A times x is a combination of
the columns of A.

613
00:38:31 --> 00:38:39
That's how I hope you will
think of A times x when we need

614
00:38:39 --> 00:38:39
it.

615
00:38:39 --> 00:38:46
Right now we've got -- with
small ones, we can always do it

616
00:38:46 --> 00:38:52
in different ways,
but later, think of it that

617
00:38:52 --> 00:38:53
way.

618
00:38:53 --> 00:38:53
Okay.

619
00:38:53 --> 00:39:01
So that's the picture for a two
by two system.

620
00:39:01 --> 00:39:07
And if the right-hand side B
happened to be twelve seven,

621
00:39:07 --> 00:39:14
then of course the correct
solution would be one two.

622
00:39:14 --> 00:39:14
Okay.

623
00:39:14 --> 00:39:20
So let me come back next time
to a systematic way,

624
00:39:20 --> 00:39:25.5
using elimination,
to find the solution,

625
00:39:25.5 --> 00:30:40
if there is one,
to a system of any size and

626
00:30:40 --> 00:17:37
find out -- because if
elimination fails,

627
00:17:37 --> 00:05:52
find out when there isn't a
solution.

628
00:05:52 --> 00:05:55
Okay, thanks.