WEBVTT
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OK, this is the lecture
on linear transformations.
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Actually, linear
algebra courses used
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to begin with this
lecture, so you
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could say I'm beginning
this course again
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by talking about
linear transformations.
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In a lot of courses, those
come first before matrices.
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The idea of a linear
transformation makes sense
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without a matrix, and
physicists and other --
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some people like
it better that way.
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They don't like coordinates.
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They don't want those numbers.
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They want to see what's going
on with the whole space.
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But, for most of us,
in the end, if we're
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going to compute anything,
we introduce coordinates,
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and then every
linear transformation
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will lead us to a
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matrix.
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And then, to all the things
that we've done about null space
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and row space, and
determinant, and eigenvalues --
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all will come from the matrix.
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But, behind it --
in other words,
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behind this is the idea of
a linear transformation.
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Let me give an example of
a linear transformation.
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So, example.
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Example one.
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A projection.
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I can describe a projection
without telling you any matrix,
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anything about any matrix.
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I can describe a
projection, say,
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this will be a linear
transformation that takes, say,
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all of R^2, every
vector in the plane,
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into a vector in the plane.
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And this is the way people
describe, a mapping.
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It takes every vector,
and so, by what rule?
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So, what's the rule, is, I
take a -- so here's the plane,
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this is going to be my line,
my line through my line,
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and I'm going to project
every vector onto that line.
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So if I take a vector like b
-- or let me call the vector v
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for the moment --
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the projection -- the linear
transformation is going
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to produce this vector as T(v).
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So T -- it's like a function.
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Exactly like a function.
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You give me an input,
the transformation
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produces the output.
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So transformation, sometimes the
word map, or mapping is used.
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A map between
inputs and outputs.
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So this is one particular
map, this is one example,
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a projection that takes
every vector -- here,
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let me do another vector v,
or let me do this vector w,
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what is T(w)?
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You see?
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There are no coordinates here.
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I've drawn those axes,
but I'm sorry I drew them,
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I'm going to remove them,
that's the whole point,
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is that we don't need axes,
we just need -- so guts --
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get it out of there,
I'm not a physicist,
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so I draw those axes.
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So the input is w, the
output of the projection
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is, project on that line, T(w).
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OK.
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Now, I could think of a
lot of transformations T.
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But, in this linear
algebra course,
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I want it to be a
linear transformation.
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So here are the rules for
a linear transformation.
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Here, see, exactly,
the two operations
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that we can do on vectors,
adding and multiplying
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by scalars, the transformation
does something special
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with respect to
those operations.
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So, for example, the projection
is a linear transformation
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because --
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for example, if I wanted
to check that one,
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if I took v to be twice
as long, the projection
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would be twice as long.
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If I took v to be minus --
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if I changed from v to
minus v, the projection
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would change to a minus.
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So c equal to two, c equal
minus one, any c is OK.
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So you see that actually, those
combine, I can combine those
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into one statement.
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What the transformation does
to any linear combination,
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it must produce the same
combination of T(v) and T(w).
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Let's think about some --
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I mean, it's like,
not hard to decide,
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is a transformation
linear or is it not.
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Let me give you an example so
you can tell me the answer.
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Suppose my transformation is
-- here's another example two.
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Shift the whole plane.
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So here are all my vectors,
my plane, and every vector
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v in the plane, I
shift it over by,
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let's say, three
by some vector v0.
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Shift whole plane by v0.
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So every vector in the plane --
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this was v, T(v) will be v+v0.
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There's T(v).
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Here's v0.
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There's the typical v.
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And there's T(v).
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You see what this
transformation does?
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Takes this vector
and adds to it.
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Adds a fixed vector to it.
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Well, that seems like
a pretty reasonable,
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simple transformation,
but is it linear?
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The answer is no,
it's not linear.
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Which law is broken?
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Maybe both laws are broken.
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Let's see.
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If I double the length of v,
does the transformation produce
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something double --
do I double T(v)?
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No.
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If I double the length of
v, in this transformation,
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I'm just adding on the same
one -- same v0, not two v0s,
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but only one v0
for every vector,
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so I don't get two
times the transform.
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Do you see what I'm saying?
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That if I double this,
then the transformation
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starts there and only goes one
v0 out and doesn't double T(v).
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In fact, a linear
transformation -- what is T of
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zero?
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That's just like a special
case, but really worth noticing.
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The zero vector in a
linear transformation
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must get transformed to zero.
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It can't move, because,
take any vector V here --
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well, so you can see
why T of zero is zero.
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Take v to be the zero
vector, take c to be three.
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Then we'd have T of
zero vector equaling
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three T of zero vector, the
T of zero has to be zero.
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OK.
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So, this example is
really a non-example.
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Shifting the whole plane is
not a linear transformation.
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Or if I cooked up some formula
that involved squaring,
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or the transformation
that, also non-example,
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how about the transformation
that, takes any vector
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and produces its length?
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So there's a transformation
that takes any vector, say,
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any vector in R^3,
let me just --
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I'll just get a chance to
use this notation again.
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Suppose I think of the
transformation that takes any
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vector in R^3 and
produces this number.
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So that, I could say, is a
member of R^1, for example,
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if I wanted.
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Or just real numbers.
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That's certainly not linear.
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It's true that the zero
vector goes to zero.
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But if I double a vector,
it does double the length,
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that's true.
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But suppose I multiply
a vector by minus two.
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What happens to its length?
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It just doubles.
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It doesn't get
multiplied by minus two.
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So when c is minus
two in my requirement,
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I'm not satisfying
that requirement.
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So T of minus v is not minus
v -- minus, the length,
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it's just the length.
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OK, so that's
another non-example.
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Projection was an example, let
me give you another example.
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I can stay here and have a --
this will be an example that is
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a linear transformation,
a rotation.
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Rotation by --
what shall we say?
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By 45 degrees.
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OK?
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So again, let me choose
this, this will be a mapping,
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from the whole plane of
vectors, into the whole plane
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of vectors, and it just --
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here is the input vector v, and
the output vector foam this 45
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degree rotation is just rotate
that thing by 45 degrees, T(v).
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So every vector got rotated.
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You see that I can describe
this without any coordinates.
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And see that it's linear.
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If I doubled v, the rotation
would just be twice as far out.
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If I had v+w, and if I rotated
each of them and added,
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the answer's the same as
if I add and then rotate.
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That's what the linear
transformation is.
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OK, so those are two examples.
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Two examples, projection
and rotation, and I
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could invent more that are
linear transformations where I
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haven't told you a matrix yet.
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Actually, the book has a
picture of the action of linear
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transformations --
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actually, the cover
of the book has it.
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So, in this section seven
point one, we can think of a --
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actually, here let's take
this linear transformation,
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rotation, suppose I have, as
the cover of the book has,
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a house in R^2.
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So instead of this, let me
take a small house in R^2.
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So that's a whole lot of points.
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The idea is, with this
linear transformation,
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that I can see what it
does to everything at once.
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I don't have to just
take one vector at a time
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and see what T of
V is, I can take
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all the vectors on the
outline of the house,
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and see where they all go.
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In fact, that will show me
where the whole house goes.
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So what will happen with
this particular linear
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transformation?
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The whole house will rotate, so
the result, if I can draw it,
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will be, the house
will be sitting there.
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OK.
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And, but suppose I give
some other examples.
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Oh, let me give some examples
that involve a matrix.
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Example three -- and
this is important --
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coming from a matrix
at -- we always call A.
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So the transformation
will be, multiply by A.
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There is a linear
transformation.
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And a whole family of
them, because every matrix
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produces a transformation
by this simple rule,
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just multiply every vector by
that matrix, and it's linear,
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right?
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Linear, I have to
check that A(v) --
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A times v plus w equals Av
plus A w, which is fine,
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and I have to check that
A times vc equals c A(v).
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Check.
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Those are fine.
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So there is a linear
transformation.
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And if I take my
favorite matrix A,
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and I apply it to all
vectors in the plane,
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it will produce a
bunch of outputs.
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See, the idea is
now worth thinking
00:14:52.020 --> 00:14:53.490
of, like, the big picture.
00:14:53.490 --> 00:15:01.670
The whole plane is transformed
by matrix multiplication.
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Every vector in the plane
gets multiplied by A.
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Let's take an example,
and see what happens
00:15:08.600 --> 00:15:10.390
to the vectors of the house.
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So this is still a
transformation from plane
00:15:13.110 --> 00:15:17.900
to plane, and let me take
a particular matrix A --
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well, if I cooked up
a rotation matrix,
00:15:22.710 --> 00:15:24.820
this would be the right picture.
00:15:24.820 --> 00:15:27.220
If I cooked up a
projection matrix,
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the projection would
be the picture.
00:15:29.550 --> 00:15:31.890
Let me just take
some other matrix.
00:15:31.890 --> 00:15:35.920
Let me take the matrix
one zero zero minus one.
00:15:39.960 --> 00:15:46.520
What happens to the house, to
all vectors, and in particular,
00:15:46.520 --> 00:15:50.150
we can sort of visualize it
if we look at the house --
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so the house is not rotated
any more, what do I get?
00:15:57.240 --> 00:16:02.160
What happens to all the vectors
if I do this transformation?
00:16:02.160 --> 00:16:04.090
I multiply by this matrix.
00:16:04.090 --> 00:16:07.660
Well, of course, it's an
easy matrix, it's diagonal.
00:16:07.660 --> 00:16:13.790
The x component stays the same,
the y component reverses sign,
00:16:13.790 --> 00:16:17.510
so that like the
roof of that house,
00:16:17.510 --> 00:16:22.890
the point, the tip of the
roof, has an x component which
00:16:22.890 --> 00:16:26.800
stays the same, but its
y component reverses,
00:16:26.800 --> 00:16:28.340
and it's down here.
00:16:28.340 --> 00:16:31.170
And, of course, what
we get is, the house
00:16:31.170 --> 00:16:33.510
is, like, upside down.
00:16:33.510 --> 00:16:36.270
Now, I have to put --
where does the door go?
00:16:36.270 --> 00:16:40.710
I guess the door goes
upside down there, right?
00:16:40.710 --> 00:16:47.280
So here's the input,
here's the input house,
00:16:47.280 --> 00:16:48.960
and this is the output.
00:16:52.610 --> 00:16:54.120
OK.
00:16:54.120 --> 00:16:57.220
This idea of a
linear transformation
00:16:57.220 --> 00:17:00.440
is like kind of the
abstract description
00:17:00.440 --> 00:17:04.030
of matrix multiplication.
00:17:04.030 --> 00:17:07.560
And what's our goal here?
00:17:07.560 --> 00:17:11.060
Our goal is to understand
linear transformations,
00:17:11.060 --> 00:17:15.200
and the way to
understand them is
00:17:15.200 --> 00:17:19.380
to find the matrix
that lies behind them.
00:17:19.380 --> 00:17:21.280
That's really the idea.
00:17:21.280 --> 00:17:23.349
Find the matrix that
lies behind them.
00:17:23.349 --> 00:17:29.500
Um, and to do that, we have
to bring in coordinates.
00:17:29.500 --> 00:17:31.630
We have to choose a basis.
00:17:31.630 --> 00:17:37.540
So let me point out
what's the story --
00:17:37.540 --> 00:17:39.810
if we have a linear
transformation --
00:17:39.810 --> 00:17:41.675
so start with --
00:17:41.675 --> 00:17:42.175
start.
00:17:48.950 --> 00:17:52.030
Suppose we have a
linear transformation.
00:17:52.030 --> 00:17:55.790
Let -- from now on, let T stand
for linear transformations.
00:17:55.790 --> 00:17:58.490
I won't be interested
in the nonlinear ones.
00:17:58.490 --> 00:18:01.110
Only linear transformations
I'm interested in.
00:18:01.110 --> 00:18:01.830
OK.
00:18:01.830 --> 00:18:05.530
I start with a linear
transformation T.
00:18:05.530 --> 00:18:11.855
Let's suppose its inputs
are vectors in R^3.
00:18:14.381 --> 00:18:14.880
OK?
00:18:14.880 --> 00:18:21.270
And suppose its outputs are
vectors in R^2, for example.
00:18:21.270 --> 00:18:22.210
OK.
00:18:22.210 --> 00:18:25.260
What's an example of
such a transformation,
00:18:25.260 --> 00:18:26.470
just before I go any further?
00:18:29.570 --> 00:18:32.930
Any matrix of the right
size will do this.
00:18:32.930 --> 00:18:35.830
So what would be the
right shape of a matrix?
00:18:35.830 --> 00:18:37.396
So, for example --
00:18:43.180 --> 00:18:44.870
I'm wanting to give
you an example,
00:18:44.870 --> 00:18:50.260
just because, here, I'm
thinking of transformations
00:18:50.260 --> 00:18:55.800
that take three-dimensional
space to two-dimensional space.
00:18:55.800 --> 00:19:01.150
And I want them to be linear,
and the easy way to invent them
00:19:01.150 --> 00:19:04.810
is a matrix multiplication.
00:19:04.810 --> 00:19:10.630
So example, T of
v should be any A
00:19:10.630 --> 00:19:13.270
v. Those transformations
are linear,
00:19:13.270 --> 00:19:15.410
that's what 18.06 is about.
00:19:15.410 --> 00:19:20.420
And A should be what size, what
shape of matrix should that be?
00:19:20.420 --> 00:19:23.310
I want V to have
three components,
00:19:23.310 --> 00:19:25.340
because this is what
the inputs have --
00:19:25.340 --> 00:19:39.220
so here's the input in R^3,
and here's the output in R^2.
00:19:39.220 --> 00:19:41.860
So what shape of matrix?
00:19:41.860 --> 00:19:50.270
So this should be, I guess,
a two by three matrix?
00:19:50.270 --> 00:19:50.770
Right?
00:19:53.860 --> 00:19:57.190
A two by three matrix.
00:19:57.190 --> 00:20:00.760
A two by three matrix, we'll
multiply a vector in R^3 --
00:20:00.760 --> 00:20:04.680
you see I'm moving to
coordinates so quickly,
00:20:04.680 --> 00:20:09.150
I'm not a true physicist here.
00:20:09.150 --> 00:20:13.060
A two by three matrix, we'll
multiply a vector in R^3
00:20:13.060 --> 00:20:16.470
an produce an output in
R^2, and it will be a linear
00:20:16.470 --> 00:20:20.660
transformation, and OK.
00:20:20.660 --> 00:20:23.840
So there's a whole
lot of examples,
00:20:23.840 --> 00:20:26.670
every two by three matrix
give me an example,
00:20:26.670 --> 00:20:29.230
and basically, I want
to show you that there
00:20:29.230 --> 00:20:30.520
are no other examples.
00:20:30.520 --> 00:20:34.540
Every linear transformation
is associated with a matrix.
00:20:34.540 --> 00:20:38.430
Now, let me come back to the
idea of linear transformation.
00:20:42.160 --> 00:20:48.030
Suppose I've got this linear
transformation in my mind,
00:20:48.030 --> 00:20:50.450
and I want to tell
you what it is.
00:20:53.220 --> 00:20:56.430
Suppose I tell you what
the transformation does
00:20:56.430 --> 00:20:58.220
to one vector.
00:20:58.220 --> 00:20:58.720
OK.
00:20:58.720 --> 00:21:00.190
You know one thing, then.
00:21:00.190 --> 00:21:01.090
All right.
00:21:01.090 --> 00:21:05.260
So this is like the -- what
I'm speaking about now is,
00:21:05.260 --> 00:21:20.980
how much information is needed
to know the transformation?
00:21:20.980 --> 00:21:24.300
By knowing T, I --
00:21:24.300 --> 00:21:28.930
to know T of v for all v.
00:21:28.930 --> 00:21:30.670
All inputs.
00:21:30.670 --> 00:21:33.520
How much information
do I have to give you
00:21:33.520 --> 00:21:35.930
so that you know what
the transformation does
00:21:35.930 --> 00:21:38.380
to every vector?
00:21:38.380 --> 00:21:40.950
OK, I could tell you what
the transformation --
00:21:40.950 --> 00:21:47.700
so I could take a vector
v1, one particular vector,
00:21:47.700 --> 00:21:53.320
tell you what the
transformation does to it --
00:21:53.320 --> 00:21:54.980
fine.
00:21:54.980 --> 00:21:57.730
But now you only know what
the transformation does to one
00:21:57.730 --> 00:21:59.170
vector.
00:21:59.170 --> 00:22:02.600
So you say, OK,
that's not enough,
00:22:02.600 --> 00:22:05.390
tell me what it does
to another vector.
00:22:05.390 --> 00:22:10.470
So I say, OK, give me a vector,
you give me a vector v2,
00:22:10.470 --> 00:22:14.805
and we see, what does the
transformation do to v2?
00:22:17.510 --> 00:22:21.570
Now, you only know --
or do you only know what
00:22:21.570 --> 00:22:23.650
the transformation
does to two vectors?
00:22:23.650 --> 00:22:27.890
Have I got to ask you --
answer you about every vector
00:22:27.890 --> 00:22:32.570
in the whole input
space, or can you,
00:22:32.570 --> 00:22:35.022
knowing what it
does to v1 and v2,
00:22:35.022 --> 00:22:37.105
how much do you now know
about the transformation?
00:22:39.730 --> 00:22:42.370
You know what the
transformation does
00:22:42.370 --> 00:22:47.860
to a larger bunch of
vectors than just these two,
00:22:47.860 --> 00:22:54.440
because you know what it does
to every linear combination.
00:22:54.440 --> 00:23:00.520
You know what it does, now,
to the whole plane of vectors,
00:23:00.520 --> 00:23:03.260
with bases v1 and v2.
00:23:03.260 --> 00:23:07.310
I'm assuming v1 and
v2 were independent.
00:23:07.310 --> 00:23:12.100
If they were dependent,
if v2 was six times v1,
00:23:12.100 --> 00:23:15.440
then I didn't give you any
new information in T of v2,
00:23:15.440 --> 00:23:20.680
you already knew it would
be six times T of v1.
00:23:20.680 --> 00:23:23.660
So you can see what
I'd headed for.
00:23:23.660 --> 00:23:27.180
If I know what the
transformation does
00:23:27.180 --> 00:23:32.150
to every vector in a basis,
then I know everything.
00:23:32.150 --> 00:23:37.110
So the information needed to
know T of v for all inputs is T
00:23:37.110 --> 00:23:47.890
of v1, T of v2, up to T
of vm, let's say, or vn,
00:23:47.890 --> 00:23:50.278
for any basis --
00:23:53.630 --> 00:23:58.280
for a basis v1 up to vn.
00:23:58.280 --> 00:24:02.030
This is a base for any --
00:24:02.030 --> 00:24:04.220
can I call it an input basis?
00:24:04.220 --> 00:24:08.330
It's a basis for
the space of inputs.
00:24:08.330 --> 00:24:11.860
The things that T is acting on.
00:24:11.860 --> 00:24:19.210
You see this point, that if
I have a basis for the input
00:24:19.210 --> 00:24:22.550
space, and I tell you what
the transformation does
00:24:22.550 --> 00:24:25.620
to every one of those
basis vectors, that
00:24:25.620 --> 00:24:31.110
is all I'm allowed to tell you,
and it's enough to know T of v
00:24:31.110 --> 00:24:33.400
for all v-s, because why?
00:24:33.400 --> 00:24:39.870
Because every v is some
combination of these basis
00:24:39.870 --> 00:24:47.430
vectors, c1v1+...+cnvn,
that's what a basis is, right?
00:24:47.430 --> 00:24:49.740
It spans the space.
00:24:49.740 --> 00:24:56.310
And if I know what T does to
this, and what T does to v2,
00:24:56.310 --> 00:25:05.220
and what T does to vn, then
I know what T does to V.
00:25:05.220 --> 00:25:13.130
By this linearity, it has
to be c1 T of v1 plus O one
00:25:13.130 --> 00:25:15.907
plus cn T of vn.
00:25:19.855 --> 00:25:20.605
There's no choice.
00:25:26.450 --> 00:25:31.380
So, the point of this
comment is that if I
00:25:31.380 --> 00:25:37.280
know what T does to a basis,
to each vector in a basis, then
00:25:37.280 --> 00:25:39.130
I know the linear
transformation.
00:25:39.130 --> 00:25:43.980
The property of linearity
tells me all the other vectors.
00:25:43.980 --> 00:25:46.420
All the other outputs.
00:25:46.420 --> 00:25:47.180
OK.
00:25:47.180 --> 00:25:54.340
So now, we got -- so
that light we now see,
00:25:54.340 --> 00:25:56.930
what do we really need in
a linear transformation,
00:25:56.930 --> 00:25:59.720
and we're ready to go to a
00:25:59.720 --> 00:26:00.740
matrix.
00:26:00.740 --> 00:26:01.470
OK.
00:26:01.470 --> 00:26:04.410
What's the step
now that takes us
00:26:04.410 --> 00:26:07.510
from a linear
transformation that's
00:26:07.510 --> 00:26:14.700
free of coordinates to a
matrix that's been created
00:26:14.700 --> 00:26:16.950
with respect to coordinates?
00:26:16.950 --> 00:26:20.190
The matrix is going to come
from the coordinate system.
00:26:20.190 --> 00:26:21.730
These are the coordinates.
00:26:21.730 --> 00:26:26.170
Coordinates mean a
basis is decided.
00:26:26.170 --> 00:26:29.570
Once you decide on a basis --
00:26:29.570 --> 00:26:30.850
this is where coordinates come
00:26:30.850 --> 00:26:31.470
from.
00:26:31.470 --> 00:26:36.280
You decide on a basis,
then every vector,
00:26:36.280 --> 00:26:41.090
these are the coordinates
in that basis.
00:26:41.090 --> 00:26:46.730
There is one and only
one way to express v
00:26:46.730 --> 00:26:49.380
as a combination of
the basis vectors,
00:26:49.380 --> 00:26:52.520
and the numbers you
need in that combination
00:26:52.520 --> 00:26:53.430
are the coordinates.
00:26:53.430 --> 00:26:55.080
Let me write that down.
00:26:55.080 --> 00:26:56.480
So what are coordinates?
00:26:56.480 --> 00:27:05.030
Coordinates come from a basis.
00:27:10.320 --> 00:27:13.690
Coordinates come from a basis.
00:27:13.690 --> 00:27:17.710
The coordinates of v,
the coordinates of v
00:27:17.710 --> 00:27:32.140
are these numbers that tell you
how much of each basis vector
00:27:32.140 --> 00:27:33.580
is in v.
00:27:33.580 --> 00:27:37.580
If I change the basis, I
change the coordinates, right?
00:27:37.580 --> 00:27:40.270
Now, we have always
been assuming
00:27:40.270 --> 00:27:44.260
that were working with
a standard basis, right?
00:27:44.260 --> 00:27:48.110
The basis we don't even
think about this stuff,
00:27:48.110 --> 00:27:55.870
because if I give you the
vector v equals three two four,
00:27:55.870 --> 00:27:59.120
you have been
assuming completely --
00:27:59.120 --> 00:28:04.050
and probably rightly -- that I
had in mind the standard basis,
00:28:04.050 --> 00:28:11.910
that this vector was three times
the first coordinate vector,
00:28:11.910 --> 00:28:16.382
and two times the second,
and four times the third.
00:28:22.130 --> 00:28:25.010
But you're not entitled --
00:28:25.010 --> 00:28:27.390
I might have had some
other basis in mind.
00:28:27.390 --> 00:28:30.520
This is like the standard basis.
00:28:30.520 --> 00:28:33.770
And then the coordinates
are sitting right there
00:28:33.770 --> 00:28:35.150
in the vector.
00:28:35.150 --> 00:28:37.210
But I could have chosen
a different basis,
00:28:37.210 --> 00:28:42.400
like I might have had
eigenvectors of a matrix,
00:28:42.400 --> 00:28:45.470
and I might have said,
OK, that's a great basis,
00:28:45.470 --> 00:28:49.110
I'll use the eigenvectors
of this matrix
00:28:49.110 --> 00:28:52.280
as my basis vectors.
00:28:52.280 --> 00:28:55.541
Which are not necessarily these
three, but some other basis.
00:28:58.130 --> 00:29:03.180
So that was an example,
this is the real thing,
00:29:03.180 --> 00:29:05.060
the coordinates
are these numbers,
00:29:05.060 --> 00:29:08.490
I'll circle them again,
the amounts of each basis.
00:29:08.490 --> 00:29:11.110
OK.
00:29:11.110 --> 00:29:15.710
So, if I want to
create a matrix that
00:29:15.710 --> 00:29:17.600
describes a linear
transformation,
00:29:17.600 --> 00:29:19.380
now I'm ready to do that.
00:29:19.380 --> 00:29:20.940
OK, OK.
00:29:20.940 --> 00:29:33.630
So now what I plan to do
is construct the matrix A
00:29:33.630 --> 00:29:42.840
that represents, or tells me
about, a linear transformation,
00:29:42.840 --> 00:29:47.230
linear transformation T. OK.
00:29:47.230 --> 00:29:51.060
So I really start with
the transformation --
00:29:51.060 --> 00:29:53.280
whether it's a
projection or a rotation,
00:29:53.280 --> 00:29:57.100
or some strange movement
of this house in the plane,
00:29:57.100 --> 00:30:02.760
or some transformation from
n-dimensional space to --
00:30:02.760 --> 00:30:05.260
or m-dimensional space
to n-dimensional space.
00:30:08.200 --> 00:30:09.840
n to m, I guess.
00:30:09.840 --> 00:30:14.990
Usually, we'll have T, we'll
somehow transform n-dimensional
00:30:14.990 --> 00:30:21.660
space to m-dimensional space,
and the whole point is that
00:30:21.660 --> 00:30:25.390
if I have a basis for
n-dimensional space --
00:30:25.390 --> 00:30:28.360
I guess I need
two bases, really.
00:30:28.360 --> 00:30:31.810
I need an input basis
to describe the inputs,
00:30:31.810 --> 00:30:36.090
and I need an output basis
to give me coordinates --
00:30:36.090 --> 00:30:39.110
to give me some
numbers for the output.
00:30:39.110 --> 00:30:41.240
So I've got to choose two bases.
00:30:41.240 --> 00:30:51.590
Choose a basis v1 up
to vn for the inputs,
00:30:51.590 --> 00:30:54.950
for the inputs in --
00:30:54.950 --> 00:30:57.550
they came from R^n.
00:30:57.550 --> 00:31:03.080
So the transformation is taking
every n-dimensional vector
00:31:03.080 --> 00:31:05.150
into some m-dimensional vector.
00:31:05.150 --> 00:31:12.520
And I have to choose a basis,
and I'll call them w1 up to wn,
00:31:12.520 --> 00:31:13.620
for the outputs.
00:31:17.120 --> 00:31:18.550
Those are guys in R^m.
00:31:22.380 --> 00:31:26.960
Once I've chosen the basis,
that settles the matrix --
00:31:26.960 --> 00:31:29.660
I now working with coordinates.
00:31:29.660 --> 00:31:35.190
Every vector in R^n, every input
vector has some coordinates.
00:31:35.190 --> 00:31:38.910
So here's what I do,
here's what I do.
00:31:38.910 --> 00:31:41.850
Can I say it in words?
00:31:41.850 --> 00:31:45.180
I take a vector v.
00:31:45.180 --> 00:31:48.214
I express it in its
basis, in the basis,
00:31:48.214 --> 00:31:49.255
so I get its coordinates.
00:31:51.820 --> 00:31:55.120
Then I'm going to multiply those
coordinates by the right matrix
00:31:55.120 --> 00:32:00.020
A, and that will give me the
coordinates of the output
00:32:00.020 --> 00:32:01.830
in the output basis.
00:32:01.830 --> 00:32:05.320
I'd better write that
down, that was a mouthful.
00:32:05.320 --> 00:32:06.870
What I want --
00:32:11.000 --> 00:32:23.080
I want a matrix A that does what
the linear transformation does.
00:32:23.080 --> 00:32:29.680
And it does it with
respecting these bases.
00:32:29.680 --> 00:32:35.000
So I want the matrix to be --
well, let's suppose -- look,
00:32:35.000 --> 00:32:37.790
let me take an example.
00:32:37.790 --> 00:32:40.150
Let me take the
projection example.
00:32:40.150 --> 00:32:42.260
The projection example.
00:32:42.260 --> 00:32:45.190
Suppose I take --
00:32:45.190 --> 00:32:47.260
because we've got that --
00:32:47.260 --> 00:32:49.240
we've got that
projection in mind --
00:32:49.240 --> 00:32:50.650
I can fit in here.
00:32:50.650 --> 00:32:52.100
Here's the projection example.
00:32:52.100 --> 00:32:58.800
So the projection example, I'm
thinking of n and m as two.
00:32:58.800 --> 00:33:01.570
The transformation
takes the plane,
00:33:01.570 --> 00:33:07.530
takes every vector in the plane,
and, let me draw the plane,
00:33:07.530 --> 00:33:10.600
just so we remember
it's a plane --
00:33:10.600 --> 00:33:15.180
and there's the thing
that I'm projecting onto,
00:33:15.180 --> 00:33:17.800
that's the line I'm
projecting onto --
00:33:17.800 --> 00:33:21.620
so the transformation takes
every vector in the plane
00:33:21.620 --> 00:33:24.360
and projects it onto that line.
00:33:24.360 --> 00:33:28.570
So this is projection, so
I'm going to do projection.
00:33:28.570 --> 00:33:29.280
OK.
00:33:29.280 --> 00:33:37.330
But, I'm going to choose
a basis that I like better
00:33:37.330 --> 00:33:39.680
than the standard basis.
00:33:39.680 --> 00:33:44.600
My basis -- in fact, I'll
choose the same basis for inputs
00:33:44.600 --> 00:33:49.260
and for outputs, and
the basis will be --
00:33:49.260 --> 00:33:53.860
my first basis vector
will be right on the line.
00:33:53.860 --> 00:33:55.330
There's my first basis vector.
00:33:55.330 --> 00:33:57.510
Say, a unit vector, on the line.
00:33:57.510 --> 00:34:01.660
And my second basis vector will
be a unit vector perpendicular
00:34:01.660 --> 00:34:02.700
to that line.
00:34:02.700 --> 00:34:05.270
And I'm going to choose
that as the output basis,
00:34:05.270 --> 00:34:06.610
also.
00:34:06.610 --> 00:34:11.010
And I'm going to ask
you, what's the matrix?
00:34:11.010 --> 00:34:14.139
What's the matrix?
00:34:14.139 --> 00:34:17.610
How do I describe this
transformation of projection
00:34:17.610 --> 00:34:20.130
with respect to this basis?
00:34:20.130 --> 00:34:20.760
OK?
00:34:20.760 --> 00:34:22.050
So what's the rule?
00:34:22.050 --> 00:34:26.110
I take any vector v,
it's some combination
00:34:26.110 --> 00:34:31.690
of the first basis ve- vector,
and the second basis vector.
00:34:31.690 --> 00:34:33.420
Now, what is T of v?
00:34:38.080 --> 00:34:45.179
Suppose the input is -- well,
suppose the input is v1.
00:34:45.179 --> 00:34:48.150
What's the output?
00:34:48.150 --> 00:34:49.870
v1, right?
00:34:49.870 --> 00:34:53.429
The projection leaves
this one alone.
00:34:53.429 --> 00:34:56.790
So we know what the projection
does to this first basis
00:34:56.790 --> 00:35:00.140
vector, this guy, it leaves it.
00:35:00.140 --> 00:35:04.240
What does the projection do
to the second basis vector?
00:35:04.240 --> 00:35:08.160
It kills it, sends it to zero.
00:35:08.160 --> 00:35:11.060
So what does the projection
do to a combination?
00:35:15.600 --> 00:35:20.060
It kills this part, and
this part, it leaves alone.
00:35:23.580 --> 00:35:27.210
Now, all I want to do
is find the matrix.
00:35:27.210 --> 00:35:29.630
I now want to find
the matrix that
00:35:29.630 --> 00:35:35.560
takes an input, c1
c2, the coordinates,
00:35:35.560 --> 00:35:39.350
and gives me the output, c1 0.
00:35:39.350 --> 00:35:44.630
You see that in this basis,
the coordinates of the input
00:35:44.630 --> 00:35:51.840
were c1, c2, and the coordinates
of the output are c1,
00:35:51.840 --> 00:35:56.590
And of course, not hard to find
a matrix that will do that.
00:35:56.590 --> 00:36:02.470
The matrix that will do that
is the matrix one, zero, zero,
00:36:02.470 --> 00:36:04.250
zero.
00:36:04.250 --> 00:36:10.560
Because if I multiply
input by that matrix A --
00:36:10.560 --> 00:36:16.130
this is A times
input coordinates --
00:36:16.130 --> 00:36:18.380
and I'm hoping to get
the output coordinates.
00:36:23.450 --> 00:36:25.610
And what do I get from
that multiplication?
00:36:25.610 --> 00:36:27.750
I get the right
answer, c1 and zero.
00:36:30.450 --> 00:36:31.970
So what's the point?
00:36:31.970 --> 00:36:36.840
So the first point is, there's
a matrix that does the job.
00:36:36.840 --> 00:36:39.250
If there's a linear
transformation out there,
00:36:39.250 --> 00:36:42.350
coordinate-free, no
coordinates, and then I
00:36:42.350 --> 00:36:45.360
choose a basis for
the inputs, and I
00:36:45.360 --> 00:36:47.790
choose a basis for
the outputs, then
00:36:47.790 --> 00:36:51.820
there's a matrix that
does And what's the job?
00:36:51.820 --> 00:36:52.570
the job.
00:36:52.570 --> 00:36:56.740
It multiplies the input
coordinates and produces
00:36:56.740 --> 00:36:58.790
the output coordinates.
00:36:58.790 --> 00:37:00.730
Now, in this example
-- let me repeat,
00:37:00.730 --> 00:37:05.380
I chose the input basis was
the same as the output basis.
00:37:05.380 --> 00:37:09.860
The input basis and output
basis were both along the line,
00:37:09.860 --> 00:37:12.510
and perpendicular to the line.
00:37:12.510 --> 00:37:16.670
They're actually the
eigenvectors of the projection.
00:37:16.670 --> 00:37:20.370
And, as a result, the
matrix came out diagonal.
00:37:20.370 --> 00:37:24.450
In fact, it came
out to be lambda.
00:37:24.450 --> 00:37:27.650
This is like, the good basis.
00:37:27.650 --> 00:37:37.640
So the good -- the eigenvector
basis is the good basis,
00:37:37.640 --> 00:37:43.210
it leads to the matrix --
00:37:43.210 --> 00:37:49.230
the diagonal matrix
of eigenvalues lambda,
00:37:49.230 --> 00:37:54.260
and just as in this example,
the eigenvectors and eigenvalues
00:37:54.260 --> 00:38:00.120
of this linear transformation
were along the line,
00:38:00.120 --> 00:38:01.630
and perpendicular.
00:38:01.630 --> 00:38:04.740
The eigenvalues
were one and zero,
00:38:04.740 --> 00:38:07.380
and that's the
matrix that we got.
00:38:07.380 --> 00:38:08.450
OK.
00:38:08.450 --> 00:38:12.290
So that's a, like, the
great choice of matrix,
00:38:12.290 --> 00:38:16.110
that's the choice a physicist
would do when he had to finally
00:38:16.110 --> 00:38:19.330
-- he or she had to
finally bring coordinates
00:38:19.330 --> 00:38:24.740
in unwillingly, the
coordinates to be chosen,
00:38:24.740 --> 00:38:27.240
the good coordinates
are the eigenvectors,
00:38:27.240 --> 00:38:32.540
because, if I did this
projection in the standard
00:38:32.540 --> 00:38:34.190
basis --
00:38:34.190 --> 00:38:35.910
which I could do, right?
00:38:35.910 --> 00:38:40.240
I could do the whole thing
in the standard basis --
00:38:40.240 --> 00:38:42.790
I better try, if I can do that.
00:38:42.790 --> 00:38:45.390
What are we calling --
00:38:45.390 --> 00:38:49.400
so I'll have to tell you now
which line we're projecting on.
00:38:49.400 --> 00:38:51.640
Say, the 45 degree line.
00:38:51.640 --> 00:38:59.200
So say we're projecting
onto 45 degree line,
00:38:59.200 --> 00:39:04.340
and we use not the eigenvector
basis, but the standard basis.
00:39:07.240 --> 00:39:15.730
The standard basis, v1, is
one, zero, and v2 is zero, one.
00:39:15.730 --> 00:39:18.720
And again, I'll use the
same basis for the outputs.
00:39:22.110 --> 00:39:24.660
Then I have to do this --
00:39:24.660 --> 00:39:29.990
I can find a matrix,
it will be the matrix
00:39:29.990 --> 00:39:31.510
that we would
always think of, it
00:39:31.510 --> 00:39:33.600
would be the projection matrix.
00:39:33.600 --> 00:39:39.610
It will be, actually, it's the
matrix that we learned about
00:39:39.610 --> 00:39:47.220
in chapter four, it's
what I call the matrix --
00:39:47.220 --> 00:39:52.920
do you remember, P was A, A
transpose over A transpose A?
00:39:52.920 --> 00:39:55.800
And I think, in this
example, it will come out,
00:39:55.800 --> 00:39:58.875
one-half, one-half,
one-half, one-half.
00:40:04.240 --> 00:40:07.940
I believe that's the matrix
that comes from our formula.
00:40:07.940 --> 00:40:10.126
And that's the matrix
that will do the job.
00:40:13.460 --> 00:40:19.020
If I give you this input,
one, zero, what's the output?
00:40:19.020 --> 00:40:20.700
The output is
one-half, one-half.
00:40:24.120 --> 00:40:28.900
And that should be
the right projection.
00:40:28.900 --> 00:40:31.010
And if I give you
the input zero, one,
00:40:31.010 --> 00:40:34.270
the output is, again, one-half,
one-half, again the projection.
00:40:37.580 --> 00:40:40.950
So that's the matrix, but
not diagonal of course,
00:40:40.950 --> 00:40:43.660
because we didn't
choose a great basis,
00:40:43.660 --> 00:40:46.700
we just chose the
handiest basis.
00:40:46.700 --> 00:40:49.040
Well, so the course
has practically
00:40:49.040 --> 00:40:53.760
been about the handiest
basis, and just dealing
00:40:53.760 --> 00:40:55.210
with the matrix that we got.
00:40:55.210 --> 00:40:59.400
And it's not that bad a
matrix, it's symmetric,
00:40:59.400 --> 00:41:02.310
and it has this P
squared equal P property,
00:41:02.310 --> 00:41:03.870
all those things are good.
00:41:03.870 --> 00:41:11.630
But in the best basis, it's easy
to see that P squared equals P,
00:41:11.630 --> 00:41:15.460
and it's symmetric,
and it's diagonal.
00:41:15.460 --> 00:41:19.480
So that's the idea
then, is, do you
00:41:19.480 --> 00:41:24.770
see now how I'm associating a
matrix to the transformation?
00:41:24.770 --> 00:41:28.910
I'd better write the rule down,
I'd better write the rule down.
00:41:28.910 --> 00:41:39.230
The rule to find the matrix A.
00:41:39.230 --> 00:41:40.370
All right, first column.
00:41:40.370 --> 00:41:50.070
So, a rule to find A,
we're given the bases.
00:41:50.070 --> 00:41:52.420
Of course, we don't -- because
there's no way we could
00:41:52.420 --> 00:41:55.390
construct the matrix until
we're told what the bases are.
00:41:55.390 --> 00:42:01.290
So we're given the input basis,
and the output basis, v1 to vn,
00:42:01.290 --> 00:42:03.660
w1 to wm.
00:42:03.660 --> 00:42:05.530
Those are given.
00:42:05.530 --> 00:42:10.550
Now, in the first column of
A, how do I find that column?
00:42:10.550 --> 00:42:13.780
The first column of the matrix.
00:42:13.780 --> 00:42:18.650
So that should tell me what
happens to the first basis
00:42:18.650 --> 00:42:19.910
vector.
00:42:19.910 --> 00:42:28.070
So the rule is, apply the
linear transformation to v1.
00:42:28.070 --> 00:42:32.280
To the first basis vector.
00:42:32.280 --> 00:42:37.010
And then, I'll write it --
so that's the output, right?
00:42:37.010 --> 00:42:40.830
The input is v1,
what's the output?
00:42:40.830 --> 00:42:42.920
The output is in
the output space,
00:42:42.920 --> 00:42:45.290
it's some combination
of these guys,
00:42:45.290 --> 00:42:50.230
and it's that combination that
goes into the first column --
00:42:50.230 --> 00:42:51.230
so, let me --
00:42:51.230 --> 00:42:57.510
I'll put this word -- right,
I'll say it in words again.
00:42:57.510 --> 00:42:59.750
How to find this matrix.
00:42:59.750 --> 00:43:01.810
Take the first basis vector.
00:43:01.810 --> 00:43:04.330
Apply the
transformation, then it's
00:43:04.330 --> 00:43:06.710
in the output space,
T of v1, so it's
00:43:06.710 --> 00:43:11.440
some combination of these
outputs, this output basis.
00:43:11.440 --> 00:43:14.870
So that combination,
the coefficients in that
00:43:14.870 --> 00:43:19.680
combination will be
the first column --
00:43:19.680 --> 00:43:32.580
so a1, a row 2,
column 1, w2, am1, wm.
00:43:32.580 --> 00:43:38.920
There are the numbers in the
first column of the matrix.
00:43:38.920 --> 00:43:41.890
Let me make the point by
doing the second column.
00:43:41.890 --> 00:43:47.960
Second column of A.
00:43:47.960 --> 00:43:49.550
What's the idea, now?
00:43:49.550 --> 00:43:54.820
I take the second basis vector,
I apply the transformation
00:43:54.820 --> 00:43:58.510
to it, that's in --
now I get an output,
00:43:58.510 --> 00:44:01.570
so it's some combination
in the output basis --
00:44:01.570 --> 00:44:06.830
and that combination is the
bunch of numbers that should go
00:44:06.830 --> 00:44:15.470
in the second column
of the matrix.
00:44:15.470 --> 00:44:16.740
OK.
00:44:16.740 --> 00:44:18.510
And so forth.
00:44:18.510 --> 00:44:22.560
So I get a matrix,
and the matrix I get
00:44:22.560 --> 00:44:24.870
does the right job.
00:44:24.870 --> 00:44:29.120
Now, the matrix constructed that
way, and following the rules
00:44:29.120 --> 00:44:31.680
of matrix multiplication.
00:44:31.680 --> 00:44:37.310
The result will be that if I
give you the input coordinates,
00:44:37.310 --> 00:44:42.810
and I multiply by the matrix,
so the outcome of all this
00:44:42.810 --> 00:44:52.370
is A times the input
coordinates correctly reproduces
00:44:52.370 --> 00:44:53.380
the output coordinates.
00:44:59.190 --> 00:45:01.520
Why is this right?
00:45:01.520 --> 00:45:04.240
Let me just check
the first column.
00:45:04.240 --> 00:45:08.697
Suppose the input coordinates
are one and all zeros.
00:45:08.697 --> 00:45:09.530
What does that mean?
00:45:09.530 --> 00:45:10.900
What's the input?
00:45:10.900 --> 00:45:13.080
If the input coordinates
are one and other --
00:45:13.080 --> 00:45:19.610
and the rest zeros, then
the input is v1, right?
00:45:19.610 --> 00:45:23.550
That's the vector that has
coordinates one and all zeros.
00:45:23.550 --> 00:45:24.270
OK?
00:45:24.270 --> 00:45:27.800
When I multiply A by
the one and all zeros,
00:45:27.800 --> 00:45:32.270
I'll get the first column of
A, I'll get these numbers.
00:45:32.270 --> 00:45:37.070
And, sure enough, those are the
output coordinates for T of v1.
00:45:37.070 --> 00:45:39.860
So we made it right
on the first column,
00:45:39.860 --> 00:45:41.940
we made it right on
the second column,
00:45:41.940 --> 00:45:44.810
we made it right on
all the basis vectors,
00:45:44.810 --> 00:45:50.380
and then it has to be
right on every vector.
00:45:50.380 --> 00:45:56.890
So there is a picture of
the matrix for a linear
00:45:56.890 --> 00:45:57.710
OK. transformation.
00:45:57.710 --> 00:46:01.610
Finally, let me
give you another --
00:46:01.610 --> 00:46:04.190
a different linear
transformation.
00:46:04.190 --> 00:46:07.485
The linear transformation
that takes the derivative.
00:46:10.110 --> 00:46:13.560
That's a linear transformation.
00:46:13.560 --> 00:46:18.590
Suppose the input space
is all combination
00:46:18.590 --> 00:46:24.510
c1 plus c2x plus c3 x squared.
00:46:24.510 --> 00:46:32.620
So the basis is these
simple functions.
00:46:32.620 --> 00:46:33.695
Then what's the output?
00:46:38.210 --> 00:46:39.770
Is the derivative.
00:46:39.770 --> 00:46:48.910
The output is the derivative,
so the output is c2+2c3 x.
00:46:48.910 --> 00:46:55.100
And let's take as output
basis, the vectors one and x.
00:46:55.100 --> 00:46:57.930
So we're going from a
three-dimensional space
00:46:57.930 --> 00:47:00.660
of inputs to a
two-dimensional space
00:47:00.660 --> 00:47:04.900
of outputs by the derivative.
00:47:04.900 --> 00:47:06.780
And I don't know
if you ever thought
00:47:06.780 --> 00:47:09.375
that the derivative is linear.
00:47:13.520 --> 00:47:16.360
But if it weren't linear,
taking derivatives
00:47:16.360 --> 00:47:19.380
would take forever, right?
00:47:19.380 --> 00:47:22.980
We are able to compute
derivatives of functions
00:47:22.980 --> 00:47:27.030
exactly because we know it's
a linear transformation,
00:47:27.030 --> 00:47:30.400
so that if we learn the
derivatives of a few functions,
00:47:30.400 --> 00:47:32.960
like sine x and cos
x and e to the x,
00:47:32.960 --> 00:47:35.730
and another little
short list, then we
00:47:35.730 --> 00:47:37.500
can take all their
combinations and we
00:47:37.500 --> 00:47:40.320
can do all the derivatives.
00:47:40.320 --> 00:47:43.010
OK, now what's the matrix?
00:47:43.010 --> 00:47:44.290
What's the matrix?
00:47:44.290 --> 00:47:50.400
So I want the matrix to
multiply these input vectors --
00:47:50.400 --> 00:47:57.200
input coordinates, and give
these output coordinates.
00:47:57.200 --> 00:48:00.250
So I just think, OK, what's
the matrix that does it?
00:48:00.250 --> 00:48:02.490
I can follow my rule
of construction,
00:48:02.490 --> 00:48:05.320
or I can see what the matrix is.
00:48:05.320 --> 00:48:10.950
It should be a two by
three matrix, right?
00:48:10.950 --> 00:48:13.230
And the matrix --
00:48:13.230 --> 00:48:15.490
so I'm just figuring
out, what do I want?
00:48:15.490 --> 00:48:17.580
No, I'll -- let
me write it here.
00:48:17.580 --> 00:48:18.995
What do I want from my matrix?
00:48:22.550 --> 00:48:23.740
What should that matrix do?
00:48:23.740 --> 00:48:26.200
Well, I want to get c2
in the first output,
00:48:26.200 --> 00:48:28.920
so zero, one, zero will do it.
00:48:28.920 --> 00:48:34.080
I want to get two c3, so
zero, zero, two will do it.
00:48:34.080 --> 00:48:40.030
That's the matrix for
this linear transformation
00:48:40.030 --> 00:48:44.440
with those bases and
those coordinates.
00:48:44.440 --> 00:48:50.330
You see, it just clicks, and the
whole point is that the inverse
00:48:50.330 --> 00:48:53.880
matrix gives the inverse to
the linear transformation,
00:48:53.880 --> 00:48:58.150
that the product of two
matrices gives the right matrix
00:48:58.150 --> 00:49:00.760
for the product of
two transformations --
00:49:00.760 --> 00:49:04.930
matrix multiplication
really came from linear
00:49:04.930 --> 00:49:05.860
transformations.
00:49:05.860 --> 00:49:10.660
I'd better pick up on that
theme Monday after Thanksgiving.
00:49:10.660 --> 00:49:13.900
And I hope you have
a great holiday.
00:49:13.900 --> 00:49:16.260
I hope Indian
summer keeps going.
00:49:16.260 --> 00:49:18.670
OK, see you on Monday.