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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
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of, like, the big picture.
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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
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to the vectors of the house.
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So this is still a
transformation from plane
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to plane, and let me take
a particular matrix A --
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well, if I cooked up
a rotation matrix,
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this would be the right picture.
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If I cooked up a
projection matrix,
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the projection would
be the picture.
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Let me just take
some other matrix.
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Let me take the matrix
one zero zero minus one.
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What happens to the house, to
all vectors, and in particular,
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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?
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What happens to all the vectors
if I do this transformation?
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I multiply by this matrix.
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Well, of course, it's an
easy matrix, it's diagonal.
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The x component stays the same,
the y component reverses sign,
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so that like the
roof of that house,
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the point, the tip of the
roof, has an x component which
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stays the same, but its
y component reverses,
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and it's down here.
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And, of course, what
we get is, the house
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is, like, upside down.
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Now, I have to put --
where does the door go?
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I guess the door goes
upside down there, right?
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So here's the input,
here's the input house,
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and this is the output.
249
00:16:52,610 --> 00:16:54,120
OK.
250
00:16:54,120 --> 00:16:57,220
This idea of a
linear transformation
251
00:16:57,220 --> 00:17:00,440
is like kind of the
abstract description
252
00:17:00,440 --> 00:17:04,030
of matrix multiplication.
253
00:17:04,030 --> 00:17:07,560
And what's our goal here?
254
00:17:07,560 --> 00:17:11,060
Our goal is to understand
linear transformations,
255
00:17:11,060 --> 00:17:15,200
and the way to
understand them is
256
00:17:15,200 --> 00:17:19,380
to find the matrix
that lies behind them.
257
00:17:19,380 --> 00:17:21,280
That's really the idea.
258
00:17:21,280 --> 00:17:23,349
Find the matrix that
lies behind them.
259
00:17:23,349 --> 00:17:29,500
Um, and to do that, we have
to bring in coordinates.
260
00:17:29,500 --> 00:17:31,630
We have to choose a basis.
261
00:17:31,630 --> 00:17:37,540
So let me point out
what's the story --
262
00:17:37,540 --> 00:17:39,810
if we have a linear
transformation --
263
00:17:39,810 --> 00:17:41,675
so start with --
264
00:17:41,675 --> 00:17:42,175
start.
265
00:17:48,950 --> 00:17:52,030
Suppose we have a
linear transformation.
266
00:17:52,030 --> 00:17:55,790
Let -- from now on, let T stand
for linear transformations.
267
00:17:55,790 --> 00:17:58,490
I won't be interested
in the nonlinear ones.
268
00:17:58,490 --> 00:18:01,110
Only linear transformations
I'm interested in.
269
00:18:01,110 --> 00:18:01,830
OK.
270
00:18:01,830 --> 00:18:05,530
I start with a linear
transformation T.
271
00:18:05,530 --> 00:18:11,855
Let's suppose its inputs
are vectors in R^3.
272
00:18:14,381 --> 00:18:14,880
OK?
273
00:18:14,880 --> 00:18:21,270
And suppose its outputs are
vectors in R^2, for example.
274
00:18:21,270 --> 00:18:22,210
OK.
275
00:18:22,210 --> 00:18:25,260
What's an example of
such a transformation,
276
00:18:25,260 --> 00:18:26,470
just before I go any further?
277
00:18:29,570 --> 00:18:32,930
Any matrix of the right
size will do this.
278
00:18:32,930 --> 00:18:35,830
So what would be the
right shape of a matrix?
279
00:18:35,830 --> 00:18:37,396
So, for example --
280
00:18:43,180 --> 00:18:44,870
I'm wanting to give
you an example,
281
00:18:44,870 --> 00:18:50,260
just because, here, I'm
thinking of transformations
282
00:18:50,260 --> 00:18:55,800
that take three-dimensional
space to two-dimensional space.
283
00:18:55,800 --> 00:19:01,150
And I want them to be linear,
and the easy way to invent them
284
00:19:01,150 --> 00:19:04,810
is a matrix multiplication.
285
00:19:04,810 --> 00:19:10,630
So example, T of
v should be any A
286
00:19:10,630 --> 00:19:13,270
v. Those transformations
are linear,
287
00:19:13,270 --> 00:19:15,410
that's what 18.06 is about.
288
00:19:15,410 --> 00:19:20,420
And A should be what size, what
shape of matrix should that be?
289
00:19:20,420 --> 00:19:23,310
I want V to have
three components,
290
00:19:23,310 --> 00:19:25,340
because this is what
the inputs have --
291
00:19:25,340 --> 00:19:39,220
so here's the input in R^3,
and here's the output in R^2.
292
00:19:39,220 --> 00:19:41,860
So what shape of matrix?
293
00:19:41,860 --> 00:19:50,270
So this should be, I guess,
a two by three matrix?
294
00:19:50,270 --> 00:19:50,770
Right?
295
00:19:53,860 --> 00:19:57,190
A two by three matrix.
296
00:19:57,190 --> 00:20:00,760
A two by three matrix, we'll
multiply a vector in R^3 --
297
00:20:00,760 --> 00:20:04,680
you see I'm moving to
coordinates so quickly,
298
00:20:04,680 --> 00:20:09,150
I'm not a true physicist here.
299
00:20:09,150 --> 00:20:13,060
A two by three matrix, we'll
multiply a vector in R^3
300
00:20:13,060 --> 00:20:16,470
an produce an output in
R^2, and it will be a linear
301
00:20:16,470 --> 00:20:20,660
transformation, and OK.
302
00:20:20,660 --> 00:20:23,840
So there's a whole
lot of examples,
303
00:20:23,840 --> 00:20:26,670
every two by three matrix
give me an example,
304
00:20:26,670 --> 00:20:29,230
and basically, I want
to show you that there
305
00:20:29,230 --> 00:20:30,520
are no other examples.
306
00:20:30,520 --> 00:20:34,540
Every linear transformation
is associated with a matrix.
307
00:20:34,540 --> 00:20:38,430
Now, let me come back to the
idea of linear transformation.
308
00:20:42,160 --> 00:20:48,030
Suppose I've got this linear
transformation in my mind,
309
00:20:48,030 --> 00:20:50,450
and I want to tell
you what it is.
310
00:20:53,220 --> 00:20:56,430
Suppose I tell you what
the transformation does
311
00:20:56,430 --> 00:20:58,220
to one vector.
312
00:20:58,220 --> 00:20:58,720
OK.
313
00:20:58,720 --> 00:21:00,190
You know one thing, then.
314
00:21:00,190 --> 00:21:01,090
All right.
315
00:21:01,090 --> 00:21:05,260
So this is like the -- what
I'm speaking about now is,
316
00:21:05,260 --> 00:21:20,980
how much information is needed
to know the transformation?
317
00:21:20,980 --> 00:21:24,300
By knowing T, I --
318
00:21:24,300 --> 00:21:28,930
to know T of v for all v.
319
00:21:28,930 --> 00:21:30,670
All inputs.
320
00:21:30,670 --> 00:21:33,520
How much information
do I have to give you
321
00:21:33,520 --> 00:21:35,930
so that you know what
the transformation does
322
00:21:35,930 --> 00:21:38,380
to every vector?
323
00:21:38,380 --> 00:21:40,950
OK, I could tell you what
the transformation --
324
00:21:40,950 --> 00:21:47,700
so I could take a vector
v1, one particular vector,
325
00:21:47,700 --> 00:21:53,320
tell you what the
transformation does to it --
326
00:21:53,320 --> 00:21:54,980
fine.
327
00:21:54,980 --> 00:21:57,730
But now you only know what
the transformation does to one
328
00:21:57,730 --> 00:21:59,170
vector.
329
00:21:59,170 --> 00:22:02,600
So you say, OK,
that's not enough,
330
00:22:02,600 --> 00:22:05,390
tell me what it does
to another vector.
331
00:22:05,390 --> 00:22:10,470
So I say, OK, give me a vector,
you give me a vector v2,
332
00:22:10,470 --> 00:22:14,805
and we see, what does the
transformation do to v2?
333
00:22:17,510 --> 00:22:21,570
Now, you only know --
or do you only know what
334
00:22:21,570 --> 00:22:23,650
the transformation
does to two vectors?
335
00:22:23,650 --> 00:22:27,890
Have I got to ask you --
answer you about every vector
336
00:22:27,890 --> 00:22:32,570
in the whole input
space, or can you,
337
00:22:32,570 --> 00:22:35,022
knowing what it
does to v1 and v2,
338
00:22:35,022 --> 00:22:37,105
how much do you now know
about the transformation?
339
00:22:39,730 --> 00:22:42,370
You know what the
transformation does
340
00:22:42,370 --> 00:22:47,860
to a larger bunch of
vectors than just these two,
341
00:22:47,860 --> 00:22:54,440
because you know what it does
to every linear combination.
342
00:22:54,440 --> 00:23:00,520
You know what it does, now,
to the whole plane of vectors,
343
00:23:00,520 --> 00:23:03,260
with bases v1 and v2.
344
00:23:03,260 --> 00:23:07,310
I'm assuming v1 and
v2 were independent.
345
00:23:07,310 --> 00:23:12,100
If they were dependent,
if v2 was six times v1,
346
00:23:12,100 --> 00:23:15,440
then I didn't give you any
new information in T of v2,
347
00:23:15,440 --> 00:23:20,680
you already knew it would
be six times T of v1.
348
00:23:20,680 --> 00:23:23,660
So you can see what
I'd headed for.
349
00:23:23,660 --> 00:23:27,180
If I know what the
transformation does
350
00:23:27,180 --> 00:23:32,150
to every vector in a basis,
then I know everything.
351
00:23:32,150 --> 00:23:37,110
So the information needed to
know T of v for all inputs is T
352
00:23:37,110 --> 00:23:47,890
of v1, T of v2, up to T
of vm, let's say, or vn,
353
00:23:47,890 --> 00:23:50,278
for any basis --
354
00:23:53,630 --> 00:23:58,280
for a basis v1 up to vn.
355
00:23:58,280 --> 00:24:02,030
This is a base for any --
356
00:24:02,030 --> 00:24:04,220
can I call it an input basis?
357
00:24:04,220 --> 00:24:08,330
It's a basis for
the space of inputs.
358
00:24:08,330 --> 00:24:11,860
The things that T is acting on.
359
00:24:11,860 --> 00:24:19,210
You see this point, that if
I have a basis for the input
360
00:24:19,210 --> 00:24:22,550
space, and I tell you what
the transformation does
361
00:24:22,550 --> 00:24:25,620
to every one of those
basis vectors, that
362
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
363
00:24:31,110 --> 00:24:33,400
for all v-s, because why?
364
00:24:33,400 --> 00:24:39,870
Because every v is some
combination of these basis
365
00:24:39,870 --> 00:24:47,430
vectors, c1v1+...+cnvn,
that's what a basis is, right?
366
00:24:47,430 --> 00:24:49,740
It spans the space.
367
00:24:49,740 --> 00:24:56,310
And if I know what T does to
this, and what T does to v2,
368
00:24:56,310 --> 00:25:05,220
and what T does to vn, then
I know what T does to V.
369
00:25:05,220 --> 00:25:13,130
By this linearity, it has
to be c1 T of v1 plus O one
370
00:25:13,130 --> 00:25:15,907
plus cn T of vn.
371
00:25:19,855 --> 00:25:20,605
There's no choice.
372
00:25:26,450 --> 00:25:31,380
So, the point of this
comment is that if I
373
00:25:31,380 --> 00:25:37,280
know what T does to a basis,
to each vector in a basis, then
374
00:25:37,280 --> 00:25:39,130
I know the linear
transformation.
375
00:25:39,130 --> 00:25:43,980
The property of linearity
tells me all the other vectors.
376
00:25:43,980 --> 00:25:46,420
All the other outputs.
377
00:25:46,420 --> 00:25:47,180
OK.
378
00:25:47,180 --> 00:25:54,340
So now, we got -- so
that light we now see,
379
00:25:54,340 --> 00:25:56,930
what do we really need in
a linear transformation,
380
00:25:56,930 --> 00:25:59,720
and we're ready to go to a
381
00:25:59,720 --> 00:26:00,740
matrix.
382
00:26:00,740 --> 00:26:01,470
OK.
383
00:26:01,470 --> 00:26:04,410
What's the step
now that takes us
384
00:26:04,410 --> 00:26:07,510
from a linear
transformation that's
385
00:26:07,510 --> 00:26:14,700
free of coordinates to a
matrix that's been created
386
00:26:14,700 --> 00:26:16,950
with respect to coordinates?
387
00:26:16,950 --> 00:26:20,190
The matrix is going to come
from the coordinate system.
388
00:26:20,190 --> 00:26:21,730
These are the coordinates.
389
00:26:21,730 --> 00:26:26,170
Coordinates mean a
basis is decided.
390
00:26:26,170 --> 00:26:29,570
Once you decide on a basis --
391
00:26:29,570 --> 00:26:30,850
this is where coordinates come
392
00:26:30,850 --> 00:26:31,470
from.
393
00:26:31,470 --> 00:26:36,280
You decide on a basis,
then every vector,
394
00:26:36,280 --> 00:26:41,090
these are the coordinates
in that basis.
395
00:26:41,090 --> 00:26:46,730
There is one and only
one way to express v
396
00:26:46,730 --> 00:26:49,380
as a combination of
the basis vectors,
397
00:26:49,380 --> 00:26:52,520
and the numbers you
need in that combination
398
00:26:52,520 --> 00:26:53,430
are the coordinates.
399
00:26:53,430 --> 00:26:55,080
Let me write that down.
400
00:26:55,080 --> 00:26:56,480
So what are coordinates?
401
00:26:56,480 --> 00:27:05,030
Coordinates come from a basis.
402
00:27:10,320 --> 00:27:13,690
Coordinates come from a basis.
403
00:27:13,690 --> 00:27:17,710
The coordinates of v,
the coordinates of v
404
00:27:17,710 --> 00:27:32,140
are these numbers that tell you
how much of each basis vector
405
00:27:32,140 --> 00:27:33,580
is in v.
406
00:27:33,580 --> 00:27:37,580
If I change the basis, I
change the coordinates, right?
407
00:27:37,580 --> 00:27:40,270
Now, we have always
been assuming
408
00:27:40,270 --> 00:27:44,260
that were working with
a standard basis, right?
409
00:27:44,260 --> 00:27:48,110
The basis we don't even
think about this stuff,
410
00:27:48,110 --> 00:27:55,870
because if I give you the
vector v equals three two four,
411
00:27:55,870 --> 00:27:59,120
you have been
assuming completely --
412
00:27:59,120 --> 00:28:04,050
and probably rightly -- that I
had in mind the standard basis,
413
00:28:04,050 --> 00:28:11,910
that this vector was three times
the first coordinate vector,
414
00:28:11,910 --> 00:28:16,382
and two times the second,
and four times the third.
415
00:28:22,130 --> 00:28:25,010
But you're not entitled --
416
00:28:25,010 --> 00:28:27,390
I might have had some
other basis in mind.
417
00:28:27,390 --> 00:28:30,520
This is like the standard basis.
418
00:28:30,520 --> 00:28:33,770
And then the coordinates
are sitting right there
419
00:28:33,770 --> 00:28:35,150
in the vector.
420
00:28:35,150 --> 00:28:37,210
But I could have chosen
a different basis,
421
00:28:37,210 --> 00:28:42,400
like I might have had
eigenvectors of a matrix,
422
00:28:42,400 --> 00:28:45,470
and I might have said,
OK, that's a great basis,
423
00:28:45,470 --> 00:28:49,110
I'll use the eigenvectors
of this matrix
424
00:28:49,110 --> 00:28:52,280
as my basis vectors.
425
00:28:52,280 --> 00:28:55,541
Which are not necessarily these
three, but some other basis.
426
00:28:58,130 --> 00:29:03,180
So that was an example,
this is the real thing,
427
00:29:03,180 --> 00:29:05,060
the coordinates
are these numbers,
428
00:29:05,060 --> 00:29:08,490
I'll circle them again,
the amounts of each basis.
429
00:29:08,490 --> 00:29:11,110
OK.
430
00:29:11,110 --> 00:29:15,710
So, if I want to
create a matrix that
431
00:29:15,710 --> 00:29:17,600
describes a linear
transformation,
432
00:29:17,600 --> 00:29:19,380
now I'm ready to do that.
433
00:29:19,380 --> 00:29:20,940
OK, OK.
434
00:29:20,940 --> 00:29:33,630
So now what I plan to do
is construct the matrix A
435
00:29:33,630 --> 00:29:42,840
that represents, or tells me
about, a linear transformation,
436
00:29:42,840 --> 00:29:47,230
linear transformation T. OK.
437
00:29:47,230 --> 00:29:51,060
So I really start with
the transformation --
438
00:29:51,060 --> 00:29:53,280
whether it's a
projection or a rotation,
439
00:29:53,280 --> 00:29:57,100
or some strange movement
of this house in the plane,
440
00:29:57,100 --> 00:30:02,760
or some transformation from
n-dimensional space to --
441
00:30:02,760 --> 00:30:05,260
or m-dimensional space
to n-dimensional space.
442
00:30:08,200 --> 00:30:09,840
n to m, I guess.
443
00:30:09,840 --> 00:30:14,990
Usually, we'll have T, we'll
somehow transform n-dimensional
444
00:30:14,990 --> 00:30:21,660
space to m-dimensional space,
and the whole point is that
445
00:30:21,660 --> 00:30:25,390
if I have a basis for
n-dimensional space --
446
00:30:25,390 --> 00:30:28,360
I guess I need
two bases, really.
447
00:30:28,360 --> 00:30:31,810
I need an input basis
to describe the inputs,
448
00:30:31,810 --> 00:30:36,090
and I need an output basis
to give me coordinates --
449
00:30:36,090 --> 00:30:39,110
to give me some
numbers for the output.
450
00:30:39,110 --> 00:30:41,240
So I've got to choose two bases.
451
00:30:41,240 --> 00:30:51,590
Choose a basis v1 up
to vn for the inputs,
452
00:30:51,590 --> 00:30:54,950
for the inputs in --
453
00:30:54,950 --> 00:30:57,550
they came from R^n.
454
00:30:57,550 --> 00:31:03,080
So the transformation is taking
every n-dimensional vector
455
00:31:03,080 --> 00:31:05,150
into some m-dimensional vector.
456
00:31:05,150 --> 00:31:12,520
And I have to choose a basis,
and I'll call them w1 up to wn,
457
00:31:12,520 --> 00:31:13,620
for the outputs.
458
00:31:17,120 --> 00:31:18,550
Those are guys in R^m.
459
00:31:22,380 --> 00:31:26,960
Once I've chosen the basis,
that settles the matrix --
460
00:31:26,960 --> 00:31:29,660
I now working with coordinates.
461
00:31:29,660 --> 00:31:35,190
Every vector in R^n, every input
vector has some coordinates.
462
00:31:35,190 --> 00:31:38,910
So here's what I do,
here's what I do.
463
00:31:38,910 --> 00:31:41,850
Can I say it in words?
464
00:31:41,850 --> 00:31:45,180
I take a vector v.
465
00:31:45,180 --> 00:31:48,214
I express it in its
basis, in the basis,
466
00:31:48,214 --> 00:31:49,255
so I get its coordinates.
467
00:31:51,820 --> 00:31:55,120
Then I'm going to multiply those
coordinates by the right matrix
468
00:31:55,120 --> 00:32:00,020
A, and that will give me the
coordinates of the output
469
00:32:00,020 --> 00:32:01,830
in the output basis.
470
00:32:01,830 --> 00:32:05,320
I'd better write that
down, that was a mouthful.
471
00:32:05,320 --> 00:32:06,870
What I want --
472
00:32:11,000 --> 00:32:23,080
I want a matrix A that does what
the linear transformation does.
473
00:32:23,080 --> 00:32:29,680
And it does it with
respecting these bases.
474
00:32:29,680 --> 00:32:35,000
So I want the matrix to be --
well, let's suppose -- look,
475
00:32:35,000 --> 00:32:37,790
let me take an example.
476
00:32:37,790 --> 00:32:40,150
Let me take the
projection example.
477
00:32:40,150 --> 00:32:42,260
The projection example.
478
00:32:42,260 --> 00:32:45,190
Suppose I take --
479
00:32:45,190 --> 00:32:47,260
because we've got that --
480
00:32:47,260 --> 00:32:49,240
we've got that
projection in mind --
481
00:32:49,240 --> 00:32:50,650
I can fit in here.
482
00:32:50,650 --> 00:32:52,100
Here's the projection example.
483
00:32:52,100 --> 00:32:58,800
So the projection example, I'm
thinking of n and m as two.
484
00:32:58,800 --> 00:33:01,570
The transformation
takes the plane,
485
00:33:01,570 --> 00:33:07,530
takes every vector in the plane,
and, let me draw the plane,
486
00:33:07,530 --> 00:33:10,600
just so we remember
it's a plane --
487
00:33:10,600 --> 00:33:15,180
and there's the thing
that I'm projecting onto,
488
00:33:15,180 --> 00:33:17,800
that's the line I'm
projecting onto --
489
00:33:17,800 --> 00:33:21,620
so the transformation takes
every vector in the plane
490
00:33:21,620 --> 00:33:24,360
and projects it onto that line.
491
00:33:24,360 --> 00:33:28,570
So this is projection, so
I'm going to do projection.
492
00:33:28,570 --> 00:33:29,280
OK.
493
00:33:29,280 --> 00:33:37,330
But, I'm going to choose
a basis that I like better
494
00:33:37,330 --> 00:33:39,680
than the standard basis.
495
00:33:39,680 --> 00:33:44,600
My basis -- in fact, I'll
choose the same basis for inputs
496
00:33:44,600 --> 00:33:49,260
and for outputs, and
the basis will be --
497
00:33:49,260 --> 00:33:53,860
my first basis vector
will be right on the line.
498
00:33:53,860 --> 00:33:55,330
There's my first basis vector.
499
00:33:55,330 --> 00:33:57,510
Say, a unit vector, on the line.
500
00:33:57,510 --> 00:34:01,660
And my second basis vector will
be a unit vector perpendicular
501
00:34:01,660 --> 00:34:02,700
to that line.
502
00:34:02,700 --> 00:34:05,270
And I'm going to choose
that as the output basis,
503
00:34:05,270 --> 00:34:06,610
also.
504
00:34:06,610 --> 00:34:11,010
And I'm going to ask
you, what's the matrix?
505
00:34:11,010 --> 00:34:14,139
What's the matrix?
506
00:34:14,139 --> 00:34:17,610
How do I describe this
transformation of projection
507
00:34:17,610 --> 00:34:20,130
with respect to this basis?
508
00:34:20,130 --> 00:34:20,760
OK?
509
00:34:20,760 --> 00:34:22,050
So what's the rule?
510
00:34:22,050 --> 00:34:26,110
I take any vector v,
it's some combination
511
00:34:26,110 --> 00:34:31,690
of the first basis ve- vector,
and the second basis vector.
512
00:34:31,690 --> 00:34:33,420
Now, what is T of v?
513
00:34:38,080 --> 00:34:45,179
Suppose the input is -- well,
suppose the input is v1.
514
00:34:45,179 --> 00:34:48,150
What's the output?
515
00:34:48,150 --> 00:34:49,870
v1, right?
516
00:34:49,870 --> 00:34:53,429
The projection leaves
this one alone.
517
00:34:53,429 --> 00:34:56,790
So we know what the projection
does to this first basis
518
00:34:56,790 --> 00:35:00,140
vector, this guy, it leaves it.
519
00:35:00,140 --> 00:35:04,240
What does the projection do
to the second basis vector?
520
00:35:04,240 --> 00:35:08,160
It kills it, sends it to zero.
521
00:35:08,160 --> 00:35:11,060
So what does the projection
do to a combination?
522
00:35:15,600 --> 00:35:20,060
It kills this part, and
this part, it leaves alone.
523
00:35:23,580 --> 00:35:27,210
Now, all I want to do
is find the matrix.
524
00:35:27,210 --> 00:35:29,630
I now want to find
the matrix that
525
00:35:29,630 --> 00:35:35,560
takes an input, c1
c2, the coordinates,
526
00:35:35,560 --> 00:35:39,350
and gives me the output, c1 0.
527
00:35:39,350 --> 00:35:44,630
You see that in this basis,
the coordinates of the input
528
00:35:44,630 --> 00:35:51,840
were c1, c2, and the coordinates
of the output are c1,
529
00:35:51,840 --> 00:35:56,590
And of course, not hard to find
a matrix that will do that.
530
00:35:56,590 --> 00:36:02,470
The matrix that will do that
is the matrix one, zero, zero,
531
00:36:02,470 --> 00:36:04,250
zero.
532
00:36:04,250 --> 00:36:10,560
Because if I multiply
input by that matrix A --
533
00:36:10,560 --> 00:36:16,130
this is A times
input coordinates --
534
00:36:16,130 --> 00:36:18,380
and I'm hoping to get
the output coordinates.
535
00:36:23,450 --> 00:36:25,610
And what do I get from
that multiplication?
536
00:36:25,610 --> 00:36:27,750
I get the right
answer, c1 and zero.
537
00:36:30,450 --> 00:36:31,970
So what's the point?
538
00:36:31,970 --> 00:36:36,840
So the first point is, there's
a matrix that does the job.
539
00:36:36,840 --> 00:36:39,250
If there's a linear
transformation out there,
540
00:36:39,250 --> 00:36:42,350
coordinate-free, no
coordinates, and then I
541
00:36:42,350 --> 00:36:45,360
choose a basis for
the inputs, and I
542
00:36:45,360 --> 00:36:47,790
choose a basis for
the outputs, then
543
00:36:47,790 --> 00:36:51,820
there's a matrix that
does And what's the job?
544
00:36:51,820 --> 00:36:52,570
the job.
545
00:36:52,570 --> 00:36:56,740
It multiplies the input
coordinates and produces
546
00:36:56,740 --> 00:36:58,790
the output coordinates.
547
00:36:58,790 --> 00:37:00,730
Now, in this example
-- let me repeat,
548
00:37:00,730 --> 00:37:05,380
I chose the input basis was
the same as the output basis.
549
00:37:05,380 --> 00:37:09,860
The input basis and output
basis were both along the line,
550
00:37:09,860 --> 00:37:12,510
and perpendicular to the line.
551
00:37:12,510 --> 00:37:16,670
They're actually the
eigenvectors of the projection.
552
00:37:16,670 --> 00:37:20,370
And, as a result, the
matrix came out diagonal.
553
00:37:20,370 --> 00:37:24,450
In fact, it came
out to be lambda.
554
00:37:24,450 --> 00:37:27,650
This is like, the good basis.
555
00:37:27,650 --> 00:37:37,640
So the good -- the eigenvector
basis is the good basis,
556
00:37:37,640 --> 00:37:43,210
it leads to the matrix --
557
00:37:43,210 --> 00:37:49,230
the diagonal matrix
of eigenvalues lambda,
558
00:37:49,230 --> 00:37:54,260
and just as in this example,
the eigenvectors and eigenvalues
559
00:37:54,260 --> 00:38:00,120
of this linear transformation
were along the line,
560
00:38:00,120 --> 00:38:01,630
and perpendicular.
561
00:38:01,630 --> 00:38:04,740
The eigenvalues
were one and zero,
562
00:38:04,740 --> 00:38:07,380
and that's the
matrix that we got.
563
00:38:07,380 --> 00:38:08,450
OK.
564
00:38:08,450 --> 00:38:12,290
So that's a, like, the
great choice of matrix,
565
00:38:12,290 --> 00:38:16,110
that's the choice a physicist
would do when he had to finally
566
00:38:16,110 --> 00:38:19,330
-- he or she had to
finally bring coordinates
567
00:38:19,330 --> 00:38:24,740
in unwillingly, the
coordinates to be chosen,
568
00:38:24,740 --> 00:38:27,240
the good coordinates
are the eigenvectors,
569
00:38:27,240 --> 00:38:32,540
because, if I did this
projection in the standard
570
00:38:32,540 --> 00:38:34,190
basis --
571
00:38:34,190 --> 00:38:35,910
which I could do, right?
572
00:38:35,910 --> 00:38:40,240
I could do the whole thing
in the standard basis --
573
00:38:40,240 --> 00:38:42,790
I better try, if I can do that.
574
00:38:42,790 --> 00:38:45,390
What are we calling --
575
00:38:45,390 --> 00:38:49,400
so I'll have to tell you now
which line we're projecting on.
576
00:38:49,400 --> 00:38:51,640
Say, the 45 degree line.
577
00:38:51,640 --> 00:38:59,200
So say we're projecting
onto 45 degree line,
578
00:38:59,200 --> 00:39:04,340
and we use not the eigenvector
basis, but the standard basis.
579
00:39:07,240 --> 00:39:15,730
The standard basis, v1, is
one, zero, and v2 is zero, one.
580
00:39:15,730 --> 00:39:18,720
And again, I'll use the
same basis for the outputs.
581
00:39:22,110 --> 00:39:24,660
Then I have to do this --
582
00:39:24,660 --> 00:39:29,990
I can find a matrix,
it will be the matrix
583
00:39:29,990 --> 00:39:31,510
that we would
always think of, it
584
00:39:31,510 --> 00:39:33,600
would be the projection matrix.
585
00:39:33,600 --> 00:39:39,610
It will be, actually, it's the
matrix that we learned about
586
00:39:39,610 --> 00:39:47,220
in chapter four, it's
what I call the matrix --
587
00:39:47,220 --> 00:39:52,920
do you remember, P was A, A
transpose over A transpose A?
588
00:39:52,920 --> 00:39:55,800
And I think, in this
example, it will come out,
589
00:39:55,800 --> 00:39:58,875
one-half, one-half,
one-half, one-half.
590
00:40:04,240 --> 00:40:07,940
I believe that's the matrix
that comes from our formula.
591
00:40:07,940 --> 00:40:10,126
And that's the matrix
that will do the job.
592
00:40:13,460 --> 00:40:19,020
If I give you this input,
one, zero, what's the output?
593
00:40:19,020 --> 00:40:20,700
The output is
one-half, one-half.
594
00:40:24,120 --> 00:40:28,900
And that should be
the right projection.
595
00:40:28,900 --> 00:40:31,010
And if I give you
the input zero, one,
596
00:40:31,010 --> 00:40:34,270
the output is, again, one-half,
one-half, again the projection.
597
00:40:37,580 --> 00:40:40,950
So that's the matrix, but
not diagonal of course,
598
00:40:40,950 --> 00:40:43,660
because we didn't
choose a great basis,
599
00:40:43,660 --> 00:40:46,700
we just chose the
handiest basis.
600
00:40:46,700 --> 00:40:49,040
Well, so the course
has practically
601
00:40:49,040 --> 00:40:53,760
been about the handiest
basis, and just dealing
602
00:40:53,760 --> 00:40:55,210
with the matrix that we got.
603
00:40:55,210 --> 00:40:59,400
And it's not that bad a
matrix, it's symmetric,
604
00:40:59,400 --> 00:41:02,310
and it has this P
squared equal P property,
605
00:41:02,310 --> 00:41:03,870
all those things are good.
606
00:41:03,870 --> 00:41:11,630
But in the best basis, it's easy
to see that P squared equals P,
607
00:41:11,630 --> 00:41:15,460
and it's symmetric,
and it's diagonal.
608
00:41:15,460 --> 00:41:19,480
So that's the idea
then, is, do you
609
00:41:19,480 --> 00:41:24,770
see now how I'm associating a
matrix to the transformation?
610
00:41:24,770 --> 00:41:28,910
I'd better write the rule down,
I'd better write the rule down.
611
00:41:28,910 --> 00:41:39,230
The rule to find the matrix A.
612
00:41:39,230 --> 00:41:40,370
All right, first column.
613
00:41:40,370 --> 00:41:50,070
So, a rule to find A,
we're given the bases.
614
00:41:50,070 --> 00:41:52,420
Of course, we don't -- because
there's no way we could
615
00:41:52,420 --> 00:41:55,390
construct the matrix until
we're told what the bases are.
616
00:41:55,390 --> 00:42:01,290
So we're given the input basis,
and the output basis, v1 to vn,
617
00:42:01,290 --> 00:42:03,660
w1 to wm.
618
00:42:03,660 --> 00:42:05,530
Those are given.
619
00:42:05,530 --> 00:42:10,550
Now, in the first column of
A, how do I find that column?
620
00:42:10,550 --> 00:42:13,780
The first column of the matrix.
621
00:42:13,780 --> 00:42:18,650
So that should tell me what
happens to the first basis
622
00:42:18,650 --> 00:42:19,910
vector.
623
00:42:19,910 --> 00:42:28,070
So the rule is, apply the
linear transformation to v1.
624
00:42:28,070 --> 00:42:32,280
To the first basis vector.
625
00:42:32,280 --> 00:42:37,010
And then, I'll write it --
so that's the output, right?
626
00:42:37,010 --> 00:42:40,830
The input is v1,
what's the output?
627
00:42:40,830 --> 00:42:42,920
The output is in
the output space,
628
00:42:42,920 --> 00:42:45,290
it's some combination
of these guys,
629
00:42:45,290 --> 00:42:50,230
and it's that combination that
goes into the first column --
630
00:42:50,230 --> 00:42:51,230
so, let me --
631
00:42:51,230 --> 00:42:57,510
I'll put this word -- right,
I'll say it in words again.
632
00:42:57,510 --> 00:42:59,750
How to find this matrix.
633
00:42:59,750 --> 00:43:01,810
Take the first basis vector.
634
00:43:01,810 --> 00:43:04,330
Apply the
transformation, then it's
635
00:43:04,330 --> 00:43:06,710
in the output space,
T of v1, so it's
636
00:43:06,710 --> 00:43:11,440
some combination of these
outputs, this output basis.
637
00:43:11,440 --> 00:43:14,870
So that combination,
the coefficients in that
638
00:43:14,870 --> 00:43:19,680
combination will be
the first column --
639
00:43:19,680 --> 00:43:32,580
so a1, a row 2,
column 1, w2, am1, wm.
640
00:43:32,580 --> 00:43:38,920
There are the numbers in the
first column of the matrix.
641
00:43:38,920 --> 00:43:41,890
Let me make the point by
doing the second column.
642
00:43:41,890 --> 00:43:47,960
Second column of A.
643
00:43:47,960 --> 00:43:49,550
What's the idea, now?
644
00:43:49,550 --> 00:43:54,820
I take the second basis vector,
I apply the transformation
645
00:43:54,820 --> 00:43:58,510
to it, that's in --
now I get an output,
646
00:43:58,510 --> 00:44:01,570
so it's some combination
in the output basis --
647
00:44:01,570 --> 00:44:06,830
and that combination is the
bunch of numbers that should go
648
00:44:06,830 --> 00:44:15,470
in the second column
of the matrix.
649
00:44:15,470 --> 00:44:16,740
OK.
650
00:44:16,740 --> 00:44:18,510
And so forth.
651
00:44:18,510 --> 00:44:22,560
So I get a matrix,
and the matrix I get
652
00:44:22,560 --> 00:44:24,870
does the right job.
653
00:44:24,870 --> 00:44:29,120
Now, the matrix constructed that
way, and following the rules
654
00:44:29,120 --> 00:44:31,680
of matrix multiplication.
655
00:44:31,680 --> 00:44:37,310
The result will be that if I
give you the input coordinates,
656
00:44:37,310 --> 00:44:42,810
and I multiply by the matrix,
so the outcome of all this
657
00:44:42,810 --> 00:44:52,370
is A times the input
coordinates correctly reproduces
658
00:44:52,370 --> 00:44:53,380
the output coordinates.
659
00:44:59,190 --> 00:45:01,520
Why is this right?
660
00:45:01,520 --> 00:45:04,240
Let me just check
the first column.
661
00:45:04,240 --> 00:45:08,697
Suppose the input coordinates
are one and all zeros.
662
00:45:08,697 --> 00:45:09,530
What does that mean?
663
00:45:09,530 --> 00:45:10,900
What's the input?
664
00:45:10,900 --> 00:45:13,080
If the input coordinates
are one and other --
665
00:45:13,080 --> 00:45:19,610
and the rest zeros, then
the input is v1, right?
666
00:45:19,610 --> 00:45:23,550
That's the vector that has
coordinates one and all zeros.
667
00:45:23,550 --> 00:45:24,270
OK?
668
00:45:24,270 --> 00:45:27,800
When I multiply A by
the one and all zeros,
669
00:45:27,800 --> 00:45:32,270
I'll get the first column of
A, I'll get these numbers.
670
00:45:32,270 --> 00:45:37,070
And, sure enough, those are the
output coordinates for T of v1.
671
00:45:37,070 --> 00:45:39,860
So we made it right
on the first column,
672
00:45:39,860 --> 00:45:41,940
we made it right on
the second column,
673
00:45:41,940 --> 00:45:44,810
we made it right on
all the basis vectors,
674
00:45:44,810 --> 00:45:50,380
and then it has to be
right on every vector.
675
00:45:50,380 --> 00:45:56,890
So there is a picture of
the matrix for a linear
676
00:45:56,890 --> 00:45:57,710
OK. transformation.
677
00:45:57,710 --> 00:46:01,610
Finally, let me
give you another --
678
00:46:01,610 --> 00:46:04,190
a different linear
transformation.
679
00:46:04,190 --> 00:46:07,485
The linear transformation
that takes the derivative.
680
00:46:10,110 --> 00:46:13,560
That's a linear transformation.
681
00:46:13,560 --> 00:46:18,590
Suppose the input space
is all combination
682
00:46:18,590 --> 00:46:24,510
c1 plus c2x plus c3 x squared.
683
00:46:24,510 --> 00:46:32,620
So the basis is these
simple functions.
684
00:46:32,620 --> 00:46:33,695
Then what's the output?
685
00:46:38,210 --> 00:46:39,770
Is the derivative.
686
00:46:39,770 --> 00:46:48,910
The output is the derivative,
so the output is c2+2c3 x.
687
00:46:48,910 --> 00:46:55,100
And let's take as output
basis, the vectors one and x.
688
00:46:55,100 --> 00:46:57,930
So we're going from a
three-dimensional space
689
00:46:57,930 --> 00:47:00,660
of inputs to a
two-dimensional space
690
00:47:00,660 --> 00:47:04,900
of outputs by the derivative.
691
00:47:04,900 --> 00:47:06,780
And I don't know
if you ever thought
692
00:47:06,780 --> 00:47:09,375
that the derivative is linear.
693
00:47:13,520 --> 00:47:16,360
But if it weren't linear,
taking derivatives
694
00:47:16,360 --> 00:47:19,380
would take forever, right?
695
00:47:19,380 --> 00:47:22,980
We are able to compute
derivatives of functions
696
00:47:22,980 --> 00:47:27,030
exactly because we know it's
a linear transformation,
697
00:47:27,030 --> 00:47:30,400
so that if we learn the
derivatives of a few functions,
698
00:47:30,400 --> 00:47:32,960
like sine x and cos
x and e to the x,
699
00:47:32,960 --> 00:47:35,730
and another little
short list, then we
700
00:47:35,730 --> 00:47:37,500
can take all their
combinations and we
701
00:47:37,500 --> 00:47:40,320
can do all the derivatives.
702
00:47:40,320 --> 00:47:43,010
OK, now what's the matrix?
703
00:47:43,010 --> 00:47:44,290
What's the matrix?
704
00:47:44,290 --> 00:47:50,400
So I want the matrix to
multiply these input vectors --
705
00:47:50,400 --> 00:47:57,200
input coordinates, and give
these output coordinates.
706
00:47:57,200 --> 00:48:00,250
So I just think, OK, what's
the matrix that does it?
707
00:48:00,250 --> 00:48:02,490
I can follow my rule
of construction,
708
00:48:02,490 --> 00:48:05,320
or I can see what the matrix is.
709
00:48:05,320 --> 00:48:10,950
It should be a two by
three matrix, right?
710
00:48:10,950 --> 00:48:13,230
And the matrix --
711
00:48:13,230 --> 00:48:15,490
so I'm just figuring
out, what do I want?
712
00:48:15,490 --> 00:48:17,580
No, I'll -- let
me write it here.
713
00:48:17,580 --> 00:48:18,995
What do I want from my matrix?
714
00:48:22,550 --> 00:48:23,740
What should that matrix do?
715
00:48:23,740 --> 00:48:26,200
Well, I want to get c2
in the first output,
716
00:48:26,200 --> 00:48:28,920
so zero, one, zero will do it.
717
00:48:28,920 --> 00:48:34,080
I want to get two c3, so
zero, zero, two will do it.
718
00:48:34,080 --> 00:48:40,030
That's the matrix for
this linear transformation
719
00:48:40,030 --> 00:48:44,440
with those bases and
those coordinates.
720
00:48:44,440 --> 00:48:50,330
You see, it just clicks, and the
whole point is that the inverse
721
00:48:50,330 --> 00:48:53,880
matrix gives the inverse to
the linear transformation,
722
00:48:53,880 --> 00:48:58,150
that the product of two
matrices gives the right matrix
723
00:48:58,150 --> 00:49:00,760
for the product of
two transformations --
724
00:49:00,760 --> 00:49:04,930
matrix multiplication
really came from linear
725
00:49:04,930 --> 00:49:05,860
transformations.
726
00:49:05,860 --> 00:49:10,660
I'd better pick up on that
theme Monday after Thanksgiving.
727
00:49:10,660 --> 00:49:13,900
And I hope you have
a great holiday.
728
00:49:13,900 --> 00:49:16,260
I hope Indian
summer keeps going.
729
00:49:16,260 --> 00:49:18,670
OK, see you on Monday.