WEBVTT
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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,
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Introduction to Linear Algebra.
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And the course web page, which
has got a lot of exercises from
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the past, MatLab codes, the
syllabus for the course,
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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
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have some number of 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, of course --
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to describe, first, what
I call the Row 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
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by two equations
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
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beside that, 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
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the 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.
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let me -- I can even
say right away --
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what's the matrix, that is,
what's the coefficient 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 --
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minus 1 in 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 --
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well, we've got two unknowns.
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So we've got a vector, with
two components, x and y,
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and we've got two right-hand
sides that go into a vector
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0 3.
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I couldn't resist writing
the matrix form, right --
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even 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 --
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I'll make that x --
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extra bold.
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A and the right-hand
side is also a vector
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that I'll always call b.
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So linear equations are A
x equal b and the idea now
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is to solve this
particular example
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and then step back to
see the bigger picture.
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Okay, what's the picture for
this example, 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
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and I'm drawing
here the xy plane
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and I'm going to plot
all the points that
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satisfy that 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, y is zero.
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The x axis has y as zero and
that -- in this case, actually,
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then x is zero.
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So the point, the origin --
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the point with 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 you
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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
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at once, because they all
lie on a straight line.
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This is a linear equation
and that word linear
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got the letters
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Okay, thanks. 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, 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 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,
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what x do we actually 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
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and I think 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
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and 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,
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connecting those
two points at --
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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,
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and this is the all-important
point that lies on both lines.
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Shall we just check
that that point which
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is the point x equal 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 equals three, okay.
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Apologies for
drawing this picture
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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,
two equations and two unknowns,
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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 --
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you see, I'm putting the two --
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I'm kind of getting the
two equations at once --
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that part and then I have a
y and in the first equation
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it's multiplying a minus one and
in the second equation a two,
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and on the right-hand
side, zero and three.
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You see, the columns of the
matrix, the columns of A
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are 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 --
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this 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 make
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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 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,
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and then I'll do a
combination of them.
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So the first column is over
two and down one, 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 --
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minus one is the first
component and 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 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
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is the right one to
produce zero three
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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
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and two 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 --
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so that's one of column
one, that's one times column
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one, 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
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vector 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,
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it would go left one and up two,
so we'd go left one and up two,
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so it would probably
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.
00:13:10.520 --> 00:13:14.350
So what I've put in here
is two of column two.
00:13:17.310 --> 00:13:19.340
Added on.
00:13:19.340 --> 00:13:20.615
And where did I end up?
00:13:23.280 --> 00:13:28.230
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?
00:13:32.570 --> 00:13:34.810
I do get that, of course.
00:13:34.810 --> 00:13:38.870
There it is, x is zero,
y is three, that's b.
00:13:38.870 --> 00:13:41.530
That's the answer we wanted.
00:13:41.530 --> 00:13:43.530
And how do I do it?
00:13:43.530 --> 00:13:46.860
You see I do it just
like the first component.
00:13:46.860 --> 00:13:50.700
I have a two and a minus
two that produces a zero,
00:13:50.700 --> 00:13:54.680
and in the second component I
have a minus one and a four,
00:13:54.680 --> 00:13:57.750
they combine to give the three.
00:13:57.750 --> 00:14:01.190
But look at this picture.
00:14:01.190 --> 00:14:03.230
So here's our key picture.
00:14:03.230 --> 00:14:10.280
I combine this column and
this column to get this guy.
00:14:10.280 --> 00:14:11.360
That was the b.
00:14:11.360 --> 00:14:13.680
That's the zero three.
00:14:13.680 --> 00:14:14.540
Okay.
00:14:14.540 --> 00:14:18.790
So that idea of linear
combination is crucial,
00:14:18.790 --> 00:14:21.130
and also --
00:14:21.130 --> 00:14:22.915
do we want to think
about this question?
00:14:25.420 --> 00:14:26.470
Sure, why not.
00:14:26.470 --> 00:14:29.830
What are all the combinations?
00:14:29.830 --> 00:14:32.655
If I took -- can I
go back to xs and ys?
00:14:35.240 --> 00:14:38.610
This is a question for really --
00:14:38.610 --> 00:14:41.450
it's going to come
up over and over,
00:14:41.450 --> 00:14:45.350
but why don't we
see it once now?
00:14:45.350 --> 00:14:50.610
If I took all the xs and all
the ys, all the combinations,
00:14:50.610 --> 00:14:53.830
what would be all the results?
00:14:53.830 --> 00:14:56.200
And, actually, the
result would be
00:14:56.200 --> 00:14:59.400
that I could get any
right-hand side at all.
00:14:59.400 --> 00:15:02.680
The combinations
of this and this
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would fill the whole plane.
00:15:06.250 --> 00:15:08.510
You can tuck that away.
00:15:08.510 --> 00:15:14.080
We'll, explore it further.
00:15:14.080 --> 00:15:19.610
But this idea of what linear
combination gives b and what do
00:15:19.610 --> 00:15:21.650
all the linear
combinations give,
00:15:21.650 --> 00:15:25.070
what are all the possible,
achievable right-hand sides be
00:15:25.070 --> 00:15:26.610
-- that's going to be basic.
00:15:26.610 --> 00:15:27.220
Okay.
00:15:27.220 --> 00:15:31.980
Can I move to three
equations and three unknowns?
00:15:31.980 --> 00:15:38.610
Because it's easy to
picture the two by two case.
00:15:38.610 --> 00:15:40.980
Let me do a three
by three example.
00:15:40.980 --> 00:15:43.330
Okay, I'll sort of
start it the same way,
00:15:43.330 --> 00:15:50.210
say maybe 2x-y and maybe I'll
take no zs as a zero and maybe
00:15:50.210 --> 00:15:55.940
a -x 2y and maybe
a -z as a -- oh,
00:15:55.940 --> 00:16:01.130
let me make that a minus one
and, just for variety let me
00:16:01.130 --> 00:16:09.930
take, -3z, -3ys, I should
keep the ys in that line,
00:16:09.930 --> 00:16:14.400
and 4zs is, say, 4.
00:16:14.400 --> 00:16:16.480
Okay.
00:16:16.480 --> 00:16:19.100
That's three equations.
00:16:19.100 --> 00:16:22.380
I'm in three
dimensions, x, y, z.
00:16:22.380 --> 00:16:27.210
And, I don't have
a solution yet.
00:16:27.210 --> 00:16:32.230
So I want to understand the
equations and then solve them.
00:16:32.230 --> 00:16:32.780
Okay.
00:16:32.780 --> 00:16:35.450
So how do I you understand them?
00:16:35.450 --> 00:16:37.350
The row picture one way.
00:16:37.350 --> 00:16:41.120
The column picture is
another very important way.
00:16:41.120 --> 00:16:43.720
Just let's remember
the matrix form, here,
00:16:43.720 --> 00:16:45.020
because that's easy.
00:16:45.020 --> 00:16:49.140
The matrix form --
what's our matrix A?
00:16:49.140 --> 00:16:54.280
Our matrix A is this right-hand
side, the two and the minus one
00:16:54.280 --> 00:16:58.620
and the zero from the first
row, the minus one and the two
00:16:58.620 --> 00:17:00.780
and the minus one
from the second row,
00:17:00.780 --> 00:17:08.530
the zero, the minus three and
the four from the third row.
00:17:08.530 --> 00:17:11.050
So it's a three by three matrix.
00:17:11.050 --> 00:17:12.970
Three equations, three unknowns.
00:17:12.970 --> 00:17:14.680
And what's our right-hand side?
00:17:14.680 --> 00:17:20.190
Of course, it's the vector,
zero minus one, four.
00:17:20.190 --> 00:17:21.180
Okay.
00:17:21.180 --> 00:17:26.950
So that's the way, well, that's
the short-hand to write out
00:17:26.950 --> 00:17:28.690
the three equations.
00:17:28.690 --> 00:17:31.429
But it's the picture that
I'm looking for today.
00:17:31.429 --> 00:17:32.470
Okay, so the row picture.
00:17:32.470 --> 00:17:38.370
All right, so I'm in
three dimensions, x,
00:17:38.370 --> 00:17:42.680
find out when there
isn't a solution.
00:17:42.680 --> 00:17:45.950
y and z.
00:17:45.950 --> 00:17:51.150
And I want to take those
equations one at a time and ask
00:17:51.150 --> 00:17:51.820
--
00:17:51.820 --> 00:17:55.280
and make a picture of all
the points that satisfy --
00:17:55.280 --> 00:17:57.960
let's take equation number two.
00:17:57.960 --> 00:18:00.820
If I make a picture of all
the points that satisfy --
00:18:00.820 --> 00:18:05.900
all the x, y, z points
that solve this equation --
00:18:05.900 --> 00:18:09.550
well, first of all, the
origin is not one of them.
00:18:09.550 --> 00:18:14.620
x, y, z -- it being 0, 0, 0
would not solve that equation.
00:18:14.620 --> 00:18:17.790
So what are some points
that do solve the equation?
00:18:17.790 --> 00:18:23.750
Let's see, maybe if x is
one, y and z could be zero.
00:18:23.750 --> 00:18:24.750
That would work, right?
00:18:24.750 --> 00:18:26.870
So there's one point.
00:18:26.870 --> 00:18:29.520
I'm looking at this
second equation,
00:18:29.520 --> 00:18:33.120
here, just, to start with.
00:18:33.120 --> 00:18:33.670
Let's see.
00:18:33.670 --> 00:18:36.430
Also, I guess, if
z could be one,
00:18:36.430 --> 00:18:38.710
x and y could be zero,
so that would just
00:18:38.710 --> 00:18:41.930
go straight up that axis.
00:18:41.930 --> 00:18:46.880
And, probably I'd want
a third point here.
00:18:46.880 --> 00:18:55.390
Let me take x to be
zero, z to be zero,
00:18:55.390 --> 00:19:00.400
then y would be
minus a half, right?
00:19:00.400 --> 00:19:06.120
So there's a third point,
somewhere -- oh my -- okay.
00:19:06.120 --> 00:19:07.900
Let's see.
00:19:07.900 --> 00:19:13.080
I want to put in all the points
that satisfy that equation.
00:19:13.080 --> 00:19:17.930
Do you know what that
bunch of points will be?
00:19:17.930 --> 00:19:19.350
It's a plane.
00:19:19.350 --> 00:19:22.430
If we have a linear
equation, then, fortunately,
00:19:22.430 --> 00:19:27.070
the graph of the thing, the plot
of all the points that solve it
00:19:27.070 --> 00:19:30.390
are a plane.
00:19:30.390 --> 00:19:32.220
These three points
determine a plane,
00:19:32.220 --> 00:19:37.570
but your lecturer
is not Rembrandt
00:19:37.570 --> 00:19:43.380
and the art is going to
be the weak point here.
00:19:43.380 --> 00:19:46.460
So I'm just going to
draw a plane, right?
00:19:46.460 --> 00:19:48.310
There's a plane somewhere.
00:19:48.310 --> 00:19:50.800
That's my plane.
00:19:50.800 --> 00:19:54.960
That plane is all the
points that solves this guy.
00:19:54.960 --> 00:19:59.880
Then, what about this one?
00:19:59.880 --> 00:20:02.760
Two x minus y plus zero z.
00:20:02.760 --> 00:20:05.410
So z actually can be anything.
00:20:05.410 --> 00:20:08.390
Again, it's going
to be another plane.
00:20:08.390 --> 00:20:11.290
Each row in a three
by three problem
00:20:11.290 --> 00:20:14.480
gives us a plane in
three dimensions.
00:20:14.480 --> 00:20:17.730
So this one is going to
be some other plane --
00:20:17.730 --> 00:20:20.250
maybe I'll try to
draw it like this.
00:20:20.250 --> 00:20:25.240
And those two planes
meet in a line.
00:20:25.240 --> 00:20:29.280
So if I have two equations,
just the first two
00:20:29.280 --> 00:20:33.380
equations in three dimensions,
those give me a line.
00:20:33.380 --> 00:20:35.290
The line where those
two planes meet.
00:20:35.290 --> 00:20:42.210
And now, the third
guy is a third plane.
00:20:42.210 --> 00:20:49.290
And it goes somewhere.
00:20:49.290 --> 00:20:51.820
Okay, those three
things meet in a point.
00:20:51.820 --> 00:20:55.090
Now I don't know where
that point is, frankly.
00:20:55.090 --> 00:20:58.140
But -- linear
algebra will find it.
00:20:58.140 --> 00:21:05.580
The main point is that the three
planes, because they're not
00:21:05.580 --> 00:21:08.100
parallel, they're not special.
00:21:08.100 --> 00:21:11.780
They do meet in one point
and that's the solution.
00:21:11.780 --> 00:21:16.910
But, maybe you can see
that this row picture is
00:21:16.910 --> 00:21:19.120
getting a little hard to see.
00:21:19.120 --> 00:21:24.520
The row picture was a cinch
when we looked at two lines
00:21:24.520 --> 00:21:25.220
meeting.
00:21:25.220 --> 00:21:27.740
When we look at
three planes meeting,
00:21:27.740 --> 00:21:32.890
it's not so clear and in
four dimensions probably
00:21:32.890 --> 00:21:34.780
a little less clear.
00:21:34.780 --> 00:21:37.530
So, can I quit on
the row picture?
00:21:37.530 --> 00:21:41.870
Or quit on the row picture
before I've successfully
00:21:41.870 --> 00:21:45.940
found the point where
the three planes meet?
00:21:45.940 --> 00:21:51.510
All I really want to see is
that the row picture consists
00:21:51.510 --> 00:21:55.780
of three planes and, if
everything works right,
00:21:55.780 --> 00:21:59.340
three planes meet in one
point and that's a solution.
00:21:59.340 --> 00:22:04.730
Now, you can tell I
prefer the column picture.
00:22:04.730 --> 00:22:06.990
Okay, so let me take
the column picture.
00:22:06.990 --> 00:22:09.820
That's x times --
00:22:09.820 --> 00:22:14.400
so there were two xs in the
first equation minus one x is,
00:22:14.400 --> 00:22:16.690
and no xs in the third.
00:22:16.690 --> 00:22:19.000
It's just the first
column of that.
00:22:19.000 --> 00:22:21.300
And how many ys are there?
00:22:21.300 --> 00:22:24.730
There's minus one in the first
equations, two in the second
00:22:24.730 --> 00:22:27.430
and maybe minus
three in the third.
00:22:27.430 --> 00:22:29.830
Just the second
column of my matrix.
00:22:29.830 --> 00:22:37.740
And z times no zs minus
one zs and four zs.
00:22:37.740 --> 00:22:41.270
And it's those three
columns, right,
00:22:41.270 --> 00:22:46.640
that I have to combine to
produce the right-hand side,
00:22:46.640 --> 00:22:48.945
which is zero minus one four.
00:22:52.830 --> 00:22:54.910
Okay.
00:22:54.910 --> 00:22:57.940
So what have we got on
this left-hand side?
00:22:57.940 --> 00:22:59.470
A linear combination.
00:22:59.470 --> 00:23:02.540
It's a linear combination
now of three vectors,
00:23:02.540 --> 00:23:05.720
and they happen to be -- each
one is a three dimensional
00:23:05.720 --> 00:23:10.000
vector, so we want to know
what combination of those three
00:23:10.000 --> 00:23:12.090
vectors produces that one.
00:23:12.090 --> 00:23:15.820
Shall I try to draw the
column picture, then?
00:23:15.820 --> 00:23:18.830
So, since these vectors
have three components --
00:23:18.830 --> 00:23:21.660
so it's some multiple -- let
me draw in the first column
00:23:21.660 --> 00:23:23.340
as before --
00:23:23.340 --> 00:23:27.287
x is two and y is minus one.
00:23:27.287 --> 00:23:28.620
Maybe there is the first column.
00:23:32.320 --> 00:23:38.570
y -- the second column has maybe
a minus one and a two and the y
00:23:38.570 --> 00:23:44.180
is a minus three, somewhere,
there possibly, column two.
00:23:44.180 --> 00:23:46.950
And the third column has --
00:23:46.950 --> 00:23:52.760
no zero minus one four,
so how shall I draw that?
00:23:52.760 --> 00:23:57.000
So this was the first component.
00:23:57.000 --> 00:23:59.230
The second component
was a minus one.
00:23:59.230 --> 00:24:01.270
Maybe up here.
00:24:01.270 --> 00:24:08.921
That's column three, that's the
column zero minus one and four.
00:24:08.921 --> 00:24:09.420
This guy.
00:24:12.050 --> 00:24:15.430
So, again, what's my problem?
00:24:15.430 --> 00:24:18.260
What this equation
is asking me to do
00:24:18.260 --> 00:24:21.340
is to combine
these three vectors
00:24:21.340 --> 00:24:27.900
with a right combination
to produce this one.
00:24:27.900 --> 00:24:33.620
Well, you can see what the
right combination is, because
00:24:33.620 --> 00:24:38.250
in this special problem,
specially chosen
00:24:38.250 --> 00:24:42.780
by the lecturer, that right-hand
side that I'm trying to get
00:24:42.780 --> 00:24:45.060
is actually one
of these columns.
00:24:45.060 --> 00:24:46.660
So I know how to get that one.
00:24:46.660 --> 00:24:48.420
So what's the solution?
00:24:48.420 --> 00:24:50.910
What combination will work?
00:24:50.910 --> 00:24:53.940
I just want one of
these and none of these.
00:24:53.940 --> 00:24:59.070
So x should be zero, y
should be zero and z should
00:24:59.070 --> 00:25:00.040
be one.
00:25:02.860 --> 00:25:04.520
That's the combination.
00:25:04.520 --> 00:25:07.600
One of those is
obviously the right one.
00:25:07.600 --> 00:25:09.870
Column three is
actually the same
00:25:09.870 --> 00:25:12.090
as b in this particular problem.
00:25:15.640 --> 00:25:17.860
I made it work
that way just so we
00:25:17.860 --> 00:25:21.770
would get an answer,
(0,0,1), so somehow that's
00:25:21.770 --> 00:25:25.890
the point where those
three planes met
00:25:25.890 --> 00:25:28.710
and I couldn't see it before.
00:25:28.710 --> 00:25:31.830
Of course, I won't always be
able to see it from the column
00:25:31.830 --> 00:25:33.170
picture, either.
00:25:33.170 --> 00:25:39.390
It's the next lecture, actually,
which is about elimination,
00:25:39.390 --> 00:25:46.670
which is the systematic
way that everybody --
00:25:46.670 --> 00:25:51.950
every bit of software, too --
00:25:51.950 --> 00:25:56.830
production, large-scale software
would solve the equations.
00:25:56.830 --> 00:25:59.020
So the lecture that's coming up.
00:25:59.020 --> 00:26:02.450
If I was to add that
to the syllabus,
00:26:02.450 --> 00:26:09.050
will be about how to find
x, y, z in all cases.
00:26:09.050 --> 00:26:15.110
Can I just think again,
though, about the big picture?
00:26:15.110 --> 00:26:18.910
By the big picture I mean
let's keep this same matrix
00:26:18.910 --> 00:26:21.990
on the left but
imagine that we have
00:26:21.990 --> 00:26:24.250
a different right-hand side.
00:26:24.250 --> 00:26:27.430
Oh, let me take a
different right-hand side.
00:26:27.430 --> 00:26:29.100
So I'll change that
right-hand side
00:26:29.100 --> 00:26:34.610
to something that actually
is also pretty special.
00:26:34.610 --> 00:26:36.810
Let me change it to --
00:26:36.810 --> 00:26:38.940
if I add those
first two columns,
00:26:38.940 --> 00:26:43.220
that would give me a one
and a one and a minus three.
00:26:43.220 --> 00:26:46.210
There's a very special
right-hand side.
00:26:46.210 --> 00:26:51.420
I just cooked it up by
adding this one to this one.
00:26:51.420 --> 00:26:54.640
Now, what's the solution with
this new right-hand side?
00:26:54.640 --> 00:26:58.960
The solution with this new
right-hand side is clear.
00:26:58.960 --> 00:27:04.880
took one of these
and none of those.
00:27:04.880 --> 00:27:08.030
So actually, it just
changed around to this
00:27:08.030 --> 00:27:11.150
when I took this
new right-hand side.
00:27:11.150 --> 00:27:12.010
Okay.
00:27:12.010 --> 00:27:19.070
So in the row picture, I
have three different planes,
00:27:19.070 --> 00:27:23.800
three new planes meeting
now at this point.
00:27:23.800 --> 00:27:27.000
In the column picture, I
have the same three columns,
00:27:27.000 --> 00:27:30.640
but now I'm combining
them to produce this guy,
00:27:30.640 --> 00:27:34.100
and it turned out that column
one plus column two which would
00:27:34.100 --> 00:27:38.500
be somewhere -- there
is the right column --
00:27:38.500 --> 00:27:42.260
one of this and one of this
would give me the new b.
00:27:45.190 --> 00:27:45.690
Okay.
00:27:45.690 --> 00:27:48.730
So we squeezed in
an extra example.
00:27:48.730 --> 00:27:55.970
But now think about all
bs, all right-hand sides.
00:27:55.970 --> 00:27:58.920
Can I solve these equations
for every right-hand side?
00:28:02.030 --> 00:28:04.980
Can I ask that question?
00:28:04.980 --> 00:28:07.020
So that's the algebra question.
00:28:07.020 --> 00:28:11.870
Can I solve A x=b for every b?
00:28:11.870 --> 00:28:13.490
Let me write that down.
00:28:13.490 --> 00:28:24.320
Can I solve A x =b for
every right-hand side b?
00:28:24.320 --> 00:28:26.420
I mean, is there a solution?
00:28:26.420 --> 00:28:29.670
And then, if there
is, elimination
00:28:29.670 --> 00:28:31.940
will give me a way to find it.
00:28:31.940 --> 00:28:34.520
I really wanted to ask,
is there a solution
00:28:34.520 --> 00:28:36.850
for every right-hand side?
00:28:36.850 --> 00:28:40.390
So now, can I put that
in different words --
00:28:40.390 --> 00:28:43.140
in this linear
combination words?
00:28:43.140 --> 00:28:52.510
So in linear combination words,
do the linear combinations
00:28:52.510 --> 00:29:03.060
of the columns fill
three dimensional space?
00:29:05.740 --> 00:29:12.500
Every b means all the bs
in three dimensional space.
00:29:12.500 --> 00:29:16.050
Do you see that I'm just
asking the same question
00:29:16.050 --> 00:29:19.600
in different words?
00:29:19.600 --> 00:29:21.040
Solving A x --
00:29:21.040 --> 00:29:25.020
A x -- that's very important.
00:29:25.020 --> 00:29:31.790
A times x -- when I multiply
a matrix by a vector,
00:29:31.790 --> 00:29:34.940
I get a combination
of the columns.
00:29:34.940 --> 00:29:38.650
I'll write that
down in a moment.
00:29:38.650 --> 00:29:43.550
But in my column picture,
that's really what I'm doing.
00:29:43.550 --> 00:29:47.080
I'm taking linear combinations
of these three columns
00:29:47.080 --> 00:29:50.620
and I'm trying to find b.
00:29:50.620 --> 00:29:57.750
And, actually, the answer
for this matrix will be yes.
00:29:57.750 --> 00:30:04.330
For this matrix A -- for these
columns, the answer is yes.
00:30:04.330 --> 00:30:19.590
This matrix -- that I chose for
an example is a good matrix.
00:30:19.590 --> 00:30:21.780
A non-singular matrix.
00:30:21.780 --> 00:30:23.180
An invertible matrix.
00:30:23.180 --> 00:30:26.700
Those will be the matrices
that we like best.
00:30:26.700 --> 00:30:29.490
There could be other --
00:30:29.490 --> 00:30:35.070
and we will see other matrices
where the answer becomes, no --
00:30:35.070 --> 00:30:38.320
oh, actually, you can see
when it would become no.
00:30:38.320 --> 00:30:43.890
What could go wrong? find out
-- because if elimination fails,
00:30:43.890 --> 00:30:46.820
How could it go wrong
that out of these --
00:30:46.820 --> 00:30:51.310
out of three columns and
all their combinations --
00:30:51.310 --> 00:30:58.390
when would I not be able
to produce some b off here?
00:30:58.390 --> 00:31:00.610
When could it go wrong?
00:31:00.610 --> 00:31:04.350
Do you see that
the combinations --
00:31:04.350 --> 00:31:06.370
let me say when it goes wrong.
00:31:06.370 --> 00:31:12.310
If these three columns
all lie in the same plane,
00:31:12.310 --> 00:31:18.110
then their combinations
will lie in that same plane.
00:31:18.110 --> 00:31:19.640
So then we're in trouble.
00:31:19.640 --> 00:31:23.880
If the three columns
of my matrix --
00:31:23.880 --> 00:31:28.180
if those three vectors happen
to lie in the same plane --
00:31:28.180 --> 00:31:31.090
for example, if
column three is just
00:31:31.090 --> 00:31:36.360
the sum of column one and column
two, I would be in trouble.
00:31:36.360 --> 00:31:40.060
That would be a matrix A
where the answer would be no,
00:31:40.060 --> 00:31:44.360
because the combinations --
00:31:44.360 --> 00:31:48.260
if column three is in the same
plane as column one and two,
00:31:48.260 --> 00:31:50.250
I don't get anything
new from that.
00:31:50.250 --> 00:31:54.610
All the combinations are in the
plane and only right-hand sides
00:31:54.610 --> 00:32:00.010
b that I could get would
be the ones in that plane.
00:32:00.010 --> 00:32:03.680
So I could solve it for
some right-hand sides, when
00:32:03.680 --> 00:32:08.320
b is in the plane, but
most right-hand sides
00:32:08.320 --> 00:32:11.270
would be out of the
plane and unreachable.
00:32:11.270 --> 00:32:14.190
So that would be
a singular case.
00:32:14.190 --> 00:32:16.830
The matrix would
be not invertible.
00:32:16.830 --> 00:32:19.810
There would not be a
solution for every b.
00:32:19.810 --> 00:32:22.511
The answer would
become no for that.
00:32:22.511 --> 00:32:23.010
Okay.
00:32:25.740 --> 00:32:27.040
I don't know --
00:32:27.040 --> 00:32:29.190
shall we take just a
little shot at thinking
00:32:29.190 --> 00:32:34.510
about nine dimensions?
00:32:34.510 --> 00:32:38.645
Imagine that we have vectors
with nine components.
00:32:41.170 --> 00:32:44.920
Well, it's going to be
hard to visualize those.
00:32:44.920 --> 00:32:46.800
I don't pretend to do it.
00:32:46.800 --> 00:32:51.950
But somehow, pretend you do.
00:32:51.950 --> 00:32:55.610
Pretend we have -- if this
was nine equations and nine
00:32:55.610 --> 00:32:59.260
unknowns, then we would
have nine columns,
00:32:59.260 --> 00:33:02.720
and each one would be a vector
in nine-dimensional space
00:33:02.720 --> 00:33:06.930
and we would be looking at
their linear combinations.
00:33:06.930 --> 00:33:09.340
So we would be having
the linear combinations
00:33:09.340 --> 00:33:12.320
of nine vectors in
nine-dimensional space,
00:33:12.320 --> 00:33:15.380
and we would be trying to
find the combination that hit
00:33:15.380 --> 00:33:18.000
the correct right-hand side b.
00:33:18.000 --> 00:33:22.990
And we might also ask the
question can we always do it?
00:33:22.990 --> 00:33:25.380
Can we get every
right-hand side b?
00:33:25.380 --> 00:33:29.840
And certainly it will depend
on those nine columns.
00:33:29.840 --> 00:33:32.480
Sometimes the answer
will be yes --
00:33:32.480 --> 00:33:35.500
if I picked a random matrix,
it would be yes, actually.
00:33:35.500 --> 00:33:39.630
If I used MatLab and just used
the random command, picked
00:33:39.630 --> 00:33:44.820
out a nine by nine matrix,
I guarantee it would be
00:33:44.820 --> 00:33:45.320
good.
00:33:45.320 --> 00:33:47.140
It would be
non-singular, it would
00:33:47.140 --> 00:33:49.270
be invertible, all beautiful.
00:33:49.270 --> 00:34:00.750
But if I choose those columns
so that they're not independent,
00:34:00.750 --> 00:34:05.790
so that the ninth column is
the same as the eighth column,
00:34:05.790 --> 00:34:09.020
then it contributes
nothing new and there
00:34:09.020 --> 00:34:13.080
would be right-hand sides
b that I couldn't get.
00:34:13.080 --> 00:34:18.210
Can you sort of think
about nine vectors
00:34:18.210 --> 00:34:22.050
in nine-dimensional space
an take their combinations?
00:34:22.050 --> 00:34:26.460
That's really the
central thought --
00:34:26.460 --> 00:34:30.920
that you get kind of used
to in linear algebra.
00:34:30.920 --> 00:34:33.340
Even though you can't
really visualize it,
00:34:33.340 --> 00:34:36.139
you sort of think you
can after a while.
00:34:36.139 --> 00:34:40.830
Those nine columns and
all their combinations
00:34:40.830 --> 00:34:45.030
may very well fill out the
whole nine-dimensional space.
00:34:45.030 --> 00:34:48.239
But if the ninth column happened
to be the same as the eighth
00:34:48.239 --> 00:34:51.090
column and gave nothing new,
then probably what it would
00:34:51.090 --> 00:34:53.102
fill out would be --
00:34:55.820 --> 00:35:03.070
I hesitate even to say this --
it would be a sort of a plane
00:35:03.070 --> 00:35:04.320
--
00:35:04.320 --> 00:35:10.030
an eight dimensional plane
inside nine-dimensional space.
00:35:10.030 --> 00:35:12.890
And it's those eight
dimensional planes
00:35:12.890 --> 00:35:16.040
inside nine-dimensional
space that we
00:35:16.040 --> 00:35:18.830
have to work with eventually.
00:35:18.830 --> 00:35:25.970
For now, let's stay with a nice
case where the matrices work,
00:35:25.970 --> 00:35:29.980
we can get every
right-hand side b and here
00:35:29.980 --> 00:35:32.350
we see how to do
it with columns.
00:35:32.350 --> 00:35:33.160
Okay.
00:35:33.160 --> 00:35:36.520
There was one step
which I realized
00:35:36.520 --> 00:35:41.210
I was saying in words that I
now want to write in letters.
00:35:41.210 --> 00:35:45.750
Because I'm coming back to the
matrix form of the equation,
00:35:45.750 --> 00:35:49.210
so let me write it here.
00:35:49.210 --> 00:35:54.270
The matrix form of my
equation, of my system
00:35:54.270 --> 00:35:57.970
is some matrix A
times some vector x
00:35:57.970 --> 00:36:00.920
equals some right-hand side b.
00:36:00.920 --> 00:36:01.950
Okay.
00:36:01.950 --> 00:36:03.450
So this is a multiplication.
00:36:03.450 --> 00:36:04.520
A times x.
00:36:04.520 --> 00:36:07.430
Matrix times vector,
and I just want to say
00:36:07.430 --> 00:36:11.620
how do you multiply
a matrix by a vector?
00:36:11.620 --> 00:36:16.550
Okay, so I'm just going
to create a matrix --
00:36:16.550 --> 00:36:21.640
let me take two
five one three --
00:36:21.640 --> 00:36:28.140
and let me take a vector
x to be, say, 1and 2.
00:36:28.140 --> 00:36:31.500
How do I multiply a
matrix by a vector?
00:36:34.150 --> 00:36:40.530
But just think a little
bit about matrix notation
00:36:40.530 --> 00:36:42.120
and how to do that
in multiplication.
00:36:42.120 --> 00:36:45.390
So let me say how I multiply
a matrix by a vector.
00:36:45.390 --> 00:36:47.760
Actually, there are
two ways to do it.
00:36:47.760 --> 00:36:50.430
Let me tell you my favorite way.
00:36:50.430 --> 00:36:52.770
It's columns again.
00:36:52.770 --> 00:36:54.720
It's a column at a time.
00:36:54.720 --> 00:36:57.980
For me, this matrix
multiplication
00:36:57.980 --> 00:37:03.290
says I take one of that column
and two of that column and add.
00:37:03.290 --> 00:37:06.770
So this is the way
I would think of it
00:37:06.770 --> 00:37:12.820
is one of the first column
and two of the second column
00:37:12.820 --> 00:37:17.300
and let's just see what we get.
00:37:17.300 --> 00:37:21.510
So in the first component
I'm getting a two and a ten.
00:37:21.510 --> 00:37:23.470
I'm getting a twelve there.
00:37:23.470 --> 00:37:26.220
In the second component I'm
getting a one and a six,
00:37:26.220 --> 00:37:27.860
I'm getting a seven.
00:37:27.860 --> 00:37:35.460
So that matrix times that
vector is twelve seven.
00:37:35.460 --> 00:37:39.207
Now, you could do
that another way.
00:37:39.207 --> 00:37:40.540
You could do it a row at a time.
00:37:40.540 --> 00:37:43.760
And you would get this twelve --
and actually I pretty much did
00:37:43.760 --> 00:37:44.970
it here --
00:37:44.970 --> 00:37:45.710
this way.
00:37:45.710 --> 00:37:48.470
Two -- I could take that
row times my vector.
00:37:48.470 --> 00:37:53.180
This is the idea
of a dot product.
00:37:53.180 --> 00:37:58.140
This vector times this vector,
two times one plus five times
00:37:58.140 --> 00:38:00.720
two is the twelve.
00:38:00.720 --> 00:38:04.610
This vector times this vector --
one times one plus three times
00:38:04.610 --> 00:38:06.000
two is the seven.
00:38:06.000 --> 00:38:11.810
So I can do it by rows,
and in each row times
00:38:11.810 --> 00:38:16.360
my x is what I'll later
call a dot product.
00:38:16.360 --> 00:38:19.430
But I also like to
see it by columns.
00:38:19.430 --> 00:38:22.490
I see this as a linear
combination of a column.
00:38:22.490 --> 00:38:24.130
So here's my point.
00:38:24.130 --> 00:38:36.180
A times x is a combination
of the columns of A.
00:38:36.180 --> 00:38:43.650
That's how I hope you will
think of A times x when we need
00:38:43.650 --> 00:38:44.150
it.
00:38:44.150 --> 00:38:47.230
Right now we've got
-- with small ones,
00:38:47.230 --> 00:38:51.390
we can always do it in
different ways, but later,
00:38:51.390 --> 00:38:53.930
think of it that way.
00:38:53.930 --> 00:38:54.480
Okay.
00:38:54.480 --> 00:39:02.020
So that's the picture
for a two by two system.
00:39:02.020 --> 00:39:05.970
And if the right-hand side B
happened to be twelve seven,
00:39:05.970 --> 00:39:12.110
then of course the correct
solution would be one two.
00:39:12.110 --> 00:39:12.610
Okay.
00:39:12.610 --> 00:39:17.950
So let me come back next
time to a systematic way,
00:39:17.950 --> 00:39:23.470
using elimination,
to find the solution,
00:39:23.470 --> 00:39:29.810
if there is one, to a
system of any size and