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OK, this is the lecture on
linear transformations.
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Actually, linear algebra
courses used to begin with this
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lecture, so you could say I'm
beginning this course again by
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talking about linear
transformations.
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In a lot of courses,
those come first before
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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 going to
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compute anything,
we introduce coordinates,
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and then every linear
transformation 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,
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and eigenvalues -- all will
come from the matrix.
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But, behind it -- in other
words, behind this is the idea
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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, this will be a linear
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transformation that takes,
say, all of R^2,
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every vector in the plane,
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
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plane, 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 for
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the moment -- the projection --
the linear transformation is
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going 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 produces the
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output.
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So transformation,
sometimes the word map,
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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 -- get
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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, I want it to be a linear
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transformation.
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So here are the rules for a
linear transformation.
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Here, see, exactly,
the two operations that we can
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do on vectors,
adding and multiplying by
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scalars, the transformation does
something special with respect
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to those operations.
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So, for example,
the projection is a linear
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transformation because --
for example,
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if I wanted to check that one,
if I took v to be twice as
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long, the projection would be
twice as long.
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If I took v to be minus -- if I
changed from v to minus v,
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the projection would change to
a minus.
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So c equal to two,
c equal minus one,
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any c is OK.
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So you see that actually,
those combine,
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I can combine those 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 --
I mean, it's like,
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not hard to decide,
is a transformation linear or
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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 v in
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the plane, I shift it over by,
let's say, three by some vector
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v0.
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Shift whole plane by v0.
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So every vector in the plane --
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, simple
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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, so I don't get two times
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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 starts
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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 must get
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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
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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 three T of zero vector,
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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 and
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produces its length?
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So there's a transformation
that takes any vector,
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say, any vector in R^3,
let me just -- I'll just get a
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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,
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for example,
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, I'm not satisfying
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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
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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 of
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vectors, and it just -- here is
the input vector v,
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and the output vector foam this
45 degree rotation is just
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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
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twice as far out.
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If I had v+w,
and if I rotated each of them
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and added, 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,
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and I could invent more that
are linear transformations where
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I 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 -- 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
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R^2.
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So that's a whole lot of
points.
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The idea is,
with this linear
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transformation,
that I can see what it does to
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everything at once.
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I don't have to just take one
vector at a time and see what T
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of V is, I can take all the
vectors on the outline of the
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house, 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,
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if I can draw it,
will be, the house will be
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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 -- coming from a
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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 produces a
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transformation by this simple
rule, just multiply every vector
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by that matrix,
and it's linear,
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right?
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Linear, I have to check that
A(v) -- A times v plus w equals
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Av plus A w, which is fine,
and I have to check that A
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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, and I apply it to all
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vectors in the plane,
it will produce a bunch of
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outputs.
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See, the idea is now worth
thinking of, like,
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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 to the
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vectors of the house.
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So this is still a
transformation from plane to
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plane, and let me take a
particular matrix A -- well,
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if I cooked up a rotation
matrix, this would be the right
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picture.
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If I cooked up a projection
matrix, the projection would be
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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,
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and in particular,
we can sort of visualize it if
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we look at the house --
so the house is not rotated any
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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,
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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, the point,
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the tip of the roof,
has an x component which stays
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the same, but its y component
reverses, and it's down here.
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And, of course,
what we get is,
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the house 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.
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OK.
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This idea of a linear
transformation is like kind of
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the abstract description of
matrix multiplication.
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And what's our goal here?
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Our goal is to understand
linear transformations,
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and the way to understand them
is to find the matrix that lies
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behind them.
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That's really the idea.
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Find the matrix that lies
behind them.
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Um, and to do that,
we have to bring in
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coordinates.
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We have to choose a basis.
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So let me point out what's the
story -- if we have a linear
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transformation -- so start with
-- start.
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Suppose we have a linear
transformation.
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Let -- from now on,
let T stand for linear
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transformations.
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I won't be interested in the
nonlinear ones.
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Only linear transformations I'm
interested in.
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OK.
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I start with a linear
transformation T.
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Let's suppose its inputs are
vectors in R^3.
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OK?
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And suppose its outputs are
vectors in R^2,
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for example.
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OK.
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What's an example of such a
transformation,
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just before I go any further?
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Any matrix of the right size
will do this.
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So what would be the right
shape of a matrix?
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So, for example -- I'm wanting
to give you an example,
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just because,
here, I'm thinking of
282
00:18:46 --> 00:18:53
transformations that take
three-dimensional space to
283
00:18:53 --> 00:18:57.74
two-dimensional space.
284
00:18:57.74 --> 00:19:03
And I want them to be linear,
and the easy way to invent them
285
00:19:03 --> 00:19:06
is a matrix multiplication.
286
00:19:06 --> 00:19:10
So example, T of v should be
any A v.
287
00:19:10 --> 00:19:15
Those transformations are
linear, that's what 18.06 is
288
00:19:15 --> 00:19:16.18
about.
289
00:19:16.18 --> 00:19:21.55
And A should be what size,
what shape of matrix should
290
00:19:21.55 --> 00:19:23
that be?
291
00:19:23 --> 00:19:30
I want V to have three
components, because this is what
292
00:19:30 --> 00:19:36
the inputs have -- so here's the
input in R^3,
293
00:19:36 --> 00:19:40
and here's the output in R^2.
294
00:19:40 --> 00:19:43
So what shape of matrix?
295
00:19:43 --> 00:19:49
So this should be,
I guess, a two by three matrix?
296
00:19:49 --> 00:19:50
Right?
297
00:19:50 --> 00:19:54
A two by three matrix.
298
00:19:54 --> 00:19:59
A two by three matrix,
we'll multiply a vector in R^3
299
00:19:59 --> 00:20:04
-- you see I'm moving to
coordinates so quickly,
300
00:20:04 --> 00:20:07.26
I'm not a true physicist here.
301
00:20:07.26 --> 00:20:12
A two by three matrix,
we'll multiply a vector in R^3
302
00:20:12 --> 00:20:17
an produce an output in R^2,
and it will be a linear
303
00:20:17 --> 00:20:20
transformation,
and OK.
304
00:20:20 --> 00:20:24
So there's a whole lot of
examples, every two by three
305
00:20:24 --> 00:20:27
matrix give me an example,
and basically,
306
00:20:27 --> 00:20:31
I want to show you that there
are no other examples.
307
00:20:31 --> 00:20:35
Every linear transformation is
associated with a matrix.
308
00:20:35 --> 00:20:40
Now, let me come back to the
idea of linear transformation.
309
00:20:40 --> 00:20:47
Suppose I've got this linear
transformation in my mind,
310
00:20:47 --> 00:20:51
and I want to tell you what it
is.
311
00:20:51 --> 00:20:58
Suppose I tell you what the
transformation does to one
312
00:20:58 --> 00:21:00
vector.
313
00:21:00 --> 00:21:00
OK.
314
00:21:00 --> 00:21:03.81
You know one thing,
then.
315
00:21:03.81 --> 00:21:05
All right.
316
00:21:05 --> 00:21:12
So this is like the -- what I'm
speaking about now is,
317
00:21:12 --> 00:21:20.59
how much information is needed
to know the transformation?
318
00:21:20.59 --> 00:21:26
By knowing T,
I -- to know T of v for all v.
319
00:21:26 --> 00:21:29
All inputs.
320
00:21:29 --> 00:21:35
How much information do I have
to give you so that you know
321
00:21:35 --> 00:21:39
what the transformation does to
every vector?
322
00:21:39 --> 00:21:44
OK, I could tell you what the
transformation -- so I could
323
00:21:44 --> 00:21:48
take a vector v1,
one particular vector,
324
00:21:48 --> 00:21:52
tell you what the
transformation does to it --
325
00:21:52 --> 00:21:54
fine.
326
00:21:54 --> 00:21:59
But now you only know what the
transformation does to one
327
00:21:59 --> 00:21:59
vector.
328
00:21:59 --> 00:22:02
So you say, OK,
that's not enough,
329
00:22:02 --> 00:22:06
tell me what it does to another
vector.
330
00:22:06 --> 00:22:08
So I say, OK,
give me a vector,
331
00:22:08 --> 00:22:13
you give me a vector v2,
and we see, what does the
332
00:22:13 --> 00:22:16
transformation do to v2?
333
00:22:16 --> 00:22:20
Now, you only know -- or do you
only know what the
334
00:22:20 --> 00:22:22
transformation does to two
vectors?
335
00:22:22 --> 00:22:27
Have I got to ask you -- answer
you about every vector in the
336
00:22:27 --> 00:22:30
whole input space,
or can you, knowing what it
337
00:22:30 --> 00:22:34
does to v1 and v2,
how much do you now know about
338
00:22:34 --> 00:22:36.98
the transformation?
339
00:22:36.98 --> 00:22:42
You know what the
transformation does to a larger
340
00:22:42 --> 00:22:49
bunch of vectors than just these
two, because you know what it
341
00:22:49 --> 00:22:53.57
does to every linear
combination.
342
00:22:53.57 --> 00:22:59
You know what it does,
now, to the whole plane of
343
00:22:59 --> 00:23:03
vectors, with bases v1 and v2.
344
00:23:03 --> 00:23:07
I'm assuming v1 and v2 were
independent.
345
00:23:07 --> 00:23:11
If they were dependent,
if v2 was six times v1,
346
00:23:11 --> 00:23:16
then I didn't give you any new
information in T of v2,
347
00:23:16 --> 00:23:20.48
you already knew it would be
six times T of v1.
348
00:23:20.48 --> 00:23:24
So you can see what I'd headed
for.
349
00:23:24 --> 00:23:30
If I know what the
transformation does to every
350
00:23:30 --> 00:23:34
vector in a basis,
then I know everything.
351
00:23:34 --> 00:23:42
So the information needed to
know T of v for all inputs is T
352
00:23:42 --> 00:23:45
of v1, T of v2,
up to T of vm,
353
00:23:45 --> 00:23:51
let's say, or vn,
for any basis -- for a basis v1
354
00:23:51 --> 00:23:53.73
up to vn.
355
00:23:53.73 --> 00:24:05
This is a base for any -- can I
call it an input basis?
356
00:24:05 --> 00:24:12
It's a basis for the space of
inputs.
357
00:24:12 --> 00:24:20
The things that T is acting on.
358
00:24:20 --> 00:24:25
You see this point,
that if I have a basis for the
359
00:24:25 --> 00:24:32
input space, and I tell you what
the transformation does to every
360
00:24:32 --> 00:24:38
one of those basis vectors,
that is all I'm allowed to tell
361
00:24:38 --> 00:24:43
you, and it's enough to know T
of v for all v-s,
362
00:24:43 --> 00:24:46
because why?
363
00:24:46 --> 00:24:51.53
Because every v is some
combination of these basis
364
00:24:51.53 --> 00:24:56.65
vectors, c1v1+...+cnvn,
that's what a basis is,
365
00:24:56.65 --> 00:24:57
right?
366
00:24:57 --> 00:24:59
It spans the space.
367
00:24:59 --> 00:25:05
And if I know what T does to
this, and what T does to v2,
368
00:25:05 --> 00:25:11
and what T does to vn,
then I know what T does to V.
369
00:25:11 --> 00:25:16
By this linearity,
it has to be c1 T of v1 plus O
370
00:25:16 --> 00:25:20
one plus cn T of vn.
371
00:25:20 --> 00:25:22
There's no choice.
372
00:25:22 --> 00:25:29
So, the point of this comment
is that if I know what T does to
373
00:25:29 --> 00:25:36
a basis, to each vector in a
basis, then I know the linear
374
00:25:36 --> 00:25:37
transformation.
375
00:25:37 --> 00:25:45
The property of linearity tells
me all the other vectors.
376
00:25:45 --> 00:25:47
All the other outputs.
377
00:25:47 --> 00:25:47
OK.
378
00:25:47 --> 00:25:52
So now, we got -- so that light
we now see, what do we really
379
00:25:52 --> 00:25:57
need in a linear transformation,
and we're ready to go to a
380
00:25:57 --> 00:25:57
matrix.
381
00:25:57 --> 00:25:58
OK.
382
00:25:58 --> 00:26:03
What's the step now that takes
us from a linear transformation
383
00:26:03 --> 00:26:08
that's free of coordinates to a
matrix that's been created with
384
00:26:08 --> 00:26:11
respect to coordinates?
385
00:26:11 --> 00:26:17
The matrix is going to come
from the coordinate system.
386
00:26:17 --> 00:26:20
These are the coordinates.
387
00:26:20 --> 00:26:23
Coordinates mean a basis is
decided.
388
00:26:23 --> 00:26:30
Once you decide on a basis --
this is where coordinates come
389
00:26:30 --> 00:26:30
from.
390
00:26:30 --> 00:26:35
You decide on a basis,
then every vector,
391
00:26:35 --> 00:26:40.5
these are the coordinates in
that basis.
392
00:26:40.5 --> 00:26:47
There is one and only one way
to express v as a combination of
393
00:26:47 --> 00:26:53
the basis vectors,
and the numbers you need in
394
00:26:53 --> 00:26:57
that combination are the
coordinates.
395
00:26:57 --> 00:27:00
Let me write that down.
396
00:27:00 --> 00:27:03
So what are coordinates?
397
00:27:03 --> 00:27:06
Coordinates come from a basis.
398
00:27:06 --> 00:27:11
Coordinates come from a basis.
399
00:27:11 --> 00:27:18
The coordinates of v,
the coordinates of v are these
400
00:27:18 --> 00:27:25
numbers that tell you how much
of each basis vector is in v.
401
00:27:25 --> 00:27:31
If I change the basis,
I change the coordinates,
402
00:27:31 --> 00:27:32
right?
403
00:27:32 --> 00:27:39
Now, we have always been
assuming that were working with
404
00:27:39 --> 00:27:43
a standard basis,
right?
405
00:27:43 --> 00:27:46
The basis we don't even think
about this stuff,
406
00:27:46 --> 00:27:51
because if I give you the
vector v equals three two four,
407
00:27:51 --> 00:27:55
you have been assuming
completely -- and probably
408
00:27:55 --> 00:27:59
rightly -- that I had in mind
the standard basis,
409
00:27:59 --> 00:28:03
that this vector was three
times the first coordinate
410
00:28:03 --> 00:28:07
vector, and two times the
second, and four times the
411
00:28:07 --> 00:28:08
third.
412
00:28:08 --> 00:28:17
But you're not entitled -- I
might have had some other basis
413
00:28:17 --> 00:28:19
in mind.
414
00:28:19 --> 00:28:23.83
This is like the standard
basis.
415
00:28:23.83 --> 00:28:32
And then the coordinates are
sitting right there in the
416
00:28:32 --> 00:28:34
vector.
417
00:28:34 --> 00:28:37
But I could have chosen a
different basis,
418
00:28:37 --> 00:28:41
like I might have had
eigenvectors of a matrix,
419
00:28:41 --> 00:28:45.42
and I might have said,
OK, that's a great basis,
420
00:28:45.42 --> 00:28:50
I'll use the eigenvectors of
this matrix as my basis vectors.
421
00:28:50 --> 00:28:56
Which are not necessarily these
three, but some other basis.
422
00:28:56 --> 00:29:00
So that was an example,
this is the real thing,
423
00:29:00 --> 00:29:05
the coordinates are these
numbers, I'll circle them again,
424
00:29:05 --> 00:29:07
the amounts of each basis.
425
00:29:07 --> 00:29:08
OK.
426
00:29:08 --> 00:29:13
So, if I want to create a
matrix that describes a linear
427
00:29:13 --> 00:29:17.91
transformation,
now I'm ready to do that.
428
00:29:17.91 --> 00:29:19
OK, OK.
429
00:29:19 --> 00:29:28
So now what I plan to do is
construct the matrix A that
430
00:29:28 --> 00:29:37
represents, or tells me about,
a linear transformation,
431
00:29:37 --> 00:29:42
linear transformation T.
432
00:29:42 --> 00:29:42
OK.
433
00:29:42 --> 00:29:51
So I really start with the
transformation --
434
00:29:51 --> 00:29:57.97
whether it's a projection or a
rotation, or some strange
435
00:29:57.97 --> 00:30:04
movement of this house in the
plane, or some transformation
436
00:30:04 --> 00:30:10.85
from n-dimensional space to --
or m-dimensional space to
437
00:30:10.85 --> 00:30:15
n-dimensional space.
n to m, I guess.
438
00:30:15 --> 00:30:19
Usually, we'll have T,
we'll somehow transform
439
00:30:19 --> 00:30:22
n-dimensional space to
m-dimensional space,
440
00:30:22 --> 00:30:26
and the whole point is that if
I have a basis for n-dimensional
441
00:30:26 --> 00:30:29.48
space -- I guess I need two
bases, really.
442
00:30:29.48 --> 00:30:32
I need an input basis to
describe the inputs,
443
00:30:32 --> 00:30:36
and I need an output basis to
give me coordinates -- to give
444
00:30:36 --> 00:30:40
me some numbers for the output.
445
00:30:40 --> 00:30:43
So I've got to choose two
bases.
446
00:30:43 --> 00:30:50
Choose a basis v1 up to vn for
the inputs, for the inputs in --
447
00:30:50 --> 00:30:53.17
they came from R^n.
448
00:30:53.17 --> 00:31:00
So the transformation is taking
every n-dimensional vector into
449
00:31:00 --> 00:31:03
some m-dimensional vector.
450
00:31:03 --> 00:31:10
And I have to choose a basis,
and I'll call them w1 up to wn,
451
00:31:10 --> 00:31:13
for the outputs.
452
00:31:13 --> 00:31:16
Those are guys in R^m.
453
00:31:16 --> 00:31:23
Once I've chosen the basis,
that settles the matrix -- I
454
00:31:23 --> 00:31:27
now working with coordinates.
455
00:31:27 --> 00:31:33
Every vector in R^n,
every input vector has some
456
00:31:33 --> 00:31:34
coordinates.
457
00:31:34 --> 00:31:39
So here's what I do,
here's what I do.
458
00:31:39 --> 00:31:43
Can I say it in words?
459
00:31:43 --> 00:31:45
I take a vector v.
460
00:31:45 --> 00:31:48
I express it in its basis,
in the basis,
461
00:31:48 --> 00:31:50
so I get its coordinates.
462
00:31:50 --> 00:31:55
Then I'm going to multiply
those coordinates by the right
463
00:31:55 --> 00:32:00
matrix A, and that will give me
the coordinates of the output in
464
00:32:00 --> 00:32:02
the output basis.
465
00:32:02 --> 00:32:07
I'd better write that down,
that was a mouthful.
466
00:32:07 --> 00:32:15
What I want -- I want a matrix
A that does what the linear
467
00:32:15 --> 00:32:18
transformation does.
468
00:32:18 --> 00:32:25
And it does it with respecting
these bases.
469
00:32:25 --> 00:32:33
So I want the matrix to be --
well, let's suppose -- look,
470
00:32:33 --> 00:32:38.5
let me take an example.
471
00:32:38.5 --> 00:32:41
Let me take the projection
example.
472
00:32:41 --> 00:32:44
The projection example.
473
00:32:44 --> 00:32:49
Suppose I take -- because we've
got that -- we've got that
474
00:32:49 --> 00:32:53
projection in mind -- I can fit
in here.
475
00:32:53 --> 00:32:56
Here's the projection example.
476
00:32:56 --> 00:33:02
So the projection example,
I'm thinking of n and m as two.
477
00:33:02 --> 00:33:08
The transformation takes the
plane, takes every vector in the
478
00:33:08 --> 00:33:13
plane, and, let me draw the
plane, just so we remember it's
479
00:33:13 --> 00:33:18
a plane -- and there's the thing
that I'm projecting onto,
480
00:33:18 --> 00:33:23
that's the line I'm projecting
onto -- so the transformation
481
00:33:23 --> 00:33:30.16
takes every vector in the plane
and projects it onto that line.
482
00:33:30.16 --> 00:33:34
So this is projection,
so I'm going to do projection.
483
00:33:34 --> 00:33:34
OK.
484
00:33:34 --> 00:33:39
But, I'm going to choose a
basis that I like better than
485
00:33:39 --> 00:33:41.22
the standard basis.
486
00:33:41.22 --> 00:33:45
My basis -- in fact,
I'll choose the same basis for
487
00:33:45 --> 00:33:49
inputs and for outputs,
and the basis will be -- my
488
00:33:49 --> 00:33:53
first basis vector will be right
on the line.
489
00:33:53 --> 00:33:57
There's my first basis vector.
490
00:33:57 --> 00:33:59
Say, a unit vector,
on the line.
491
00:33:59 --> 00:34:03
And my second basis vector will
be a unit vector perpendicular
492
00:34:03 --> 00:34:04
to that line.
493
00:34:04 --> 00:34:08
And I'm going to choose that as
the output basis,
494
00:34:08 --> 00:34:08
also.
495
00:34:08 --> 00:34:11
And I'm going to ask you,
what's the matrix?
496
00:34:11 --> 00:34:13
What's the matrix?
497
00:34:13 --> 00:34:16
How do I describe this
transformation of projection
498
00:34:16 --> 00:34:19
with respect to this basis?
499
00:34:19 --> 00:34:20
OK?
500
00:34:20 --> 00:34:22
So what's the rule?
501
00:34:22 --> 00:34:28.53
I take any vector v,
it's some combination of the
502
00:34:28.53 --> 00:34:34
first basis ve- vector,
and the second basis vector.
503
00:34:34 --> 00:34:37
Now, what is T of v?
504
00:34:37 --> 00:34:43
Suppose the input is -- well,
suppose the input is v1.
505
00:34:43 --> 00:34:48
What's the output?
v1, right?
506
00:34:48 --> 00:34:51
The projection leaves this one
alone.
507
00:34:51 --> 00:34:57
So we know what the projection
does to this first basis vector,
508
00:34:57 --> 00:34:59
this guy, it leaves it.
509
00:34:59 --> 00:35:04
What does the projection do to
the second basis vector?
510
00:35:04 --> 00:35:07
It kills it,
sends it to zero.
511
00:35:07 --> 00:35:11
So what does the projection do
to a combination?
512
00:35:11 --> 00:35:14
It kills this part,
and this part,
513
00:35:14 --> 00:35:17
it leaves alone.
514
00:35:17 --> 00:35:23
Now, all I want to do is find
the matrix.
515
00:35:23 --> 00:35:30
I now want to find the matrix
that takes an input,
516
00:35:30 --> 00:35:37
c1 c2, the coordinates,
and gives me the output,
517
00:35:37 --> 00:35:39
c1 0.
518
00:35:39 --> 00:35:44
You see that in this basis,
the coordinates of the input
519
00:35:44 --> 00:35:49.33
were c1, c2, and the coordinates
of the output are c1,
520
00:35:49.33 --> 0.
521
0. --> 00:35:49
522
00:35:49 --> 00:35:53
And of course,
not hard to find a matrix that
523
00:35:53 --> 00:35:54
will do that.
524
00:35:54 --> 00:35:58
The matrix that will do that is
the matrix one,
525
00:35:58 --> 00:36:01
zero, zero, zero.
526
00:36:01 --> 00:36:09
Because if I multiply input by
that matrix A -- this is A times
527
00:36:09 --> 00:36:16
input coordinates -- and I'm
hoping to get the output
528
00:36:16 --> 00:36:18
coordinates.
529
00:36:18 --> 00:36:23
And what do I get from that
multiplication?
530
00:36:23 --> 00:36:28.37
I get the right answer,
c1 and zero.
531
00:36:28.37 --> 00:36:32
So what's the point?
532
00:36:32 --> 00:36:35
So the first point is,
there's a matrix that does the
533
00:36:35 --> 00:36:36
job.
534
00:36:36 --> 00:36:39
If there's a linear
transformation out there,
535
00:36:39 --> 00:36:41
coordinate-free,
no coordinates,
536
00:36:41 --> 00:36:45
and then I choose a basis for
the inputs, and I choose a basis
537
00:36:45 --> 00:36:48.44
for the outputs,
then there's a matrix that does
538
00:36:48.44 --> 00:36:48
the job.
539
00:36:48 --> 00:36:50
And what's the job?
540
00:36:50 --> 00:36:53
It multiplies the input
coordinates and produces the
541
00:36:53 --> 00:36:56
output coordinates.
542
00:36:56 --> 00:37:01
Now, in this example -- let me
repeat, I chose the input basis
543
00:37:01 --> 00:37:04
was the same as the output
basis.
544
00:37:04 --> 00:37:08
The input basis and output
basis were both along the line,
545
00:37:08 --> 00:37:11
and perpendicular to the line.
546
00:37:11 --> 00:37:16
They're actually the
eigenvectors of the projection.
547
00:37:16 --> 00:37:22
And, as a result,
the matrix came out diagonal.
548
00:37:22 --> 00:37:27
In fact, it came out to be
lambda.
549
00:37:27 --> 00:37:31
This is like,
the good basis.
550
00:37:31 --> 00:37:38
So the good -- the eigenvector
basis is the good basis,
551
00:37:38 --> 00:37:45
it leads to the matrix --
the diagonal matrix of
552
00:37:45 --> 00:37:50
eigenvalues lambda,
and just as in this example,
553
00:37:50 --> 00:37:56
the eigenvectors and
eigenvalues of this linear
554
00:37:56 --> 00:38:01
transformation were along the
line, and perpendicular.
555
00:38:01 --> 00:38:08
The eigenvalues were one and
zero, and that's the matrix that
556
00:38:08 --> 00:38:09
we got.
557
00:38:09 --> 00:38:10
OK.
558
00:38:10 --> 00:38:13
So that's a,
like, the great choice of
559
00:38:13 --> 00:38:17
matrix, that's the choice a
physicist would do when he had
560
00:38:17 --> 00:38:21
to finally -- he or she had to
finally bring coordinates in
561
00:38:21 --> 00:38:25
unwillingly, the coordinates to
be chosen, the good coordinates
562
00:38:25 --> 00:38:28
are the eigenvectors,
because, if I did this
563
00:38:28 --> 00:38:32
projection in the standard basis
-- which I could do,
564
00:38:32 --> 00:38:33
right?
565
00:38:33 --> 00:38:40.46
I could do the whole thing in
the standard basis -- I better
566
00:38:40.46 --> 00:38:43
try, if I can do that.
567
00:38:43 --> 00:38:50
What are we calling -- so I'll
have to tell you now which line
568
00:38:50 --> 00:38:53
we're projecting on.
569
00:38:53 --> 00:38:56
Say, the 45 degree line.
570
00:38:56 --> 00:39:03
So say we're projecting onto 45
degree line, and we use not the
571
00:39:03 --> 00:39:08
eigenvector basis,
but the standard basis.
572
00:39:08 --> 00:39:12
The standard basis,
v1, is one, zero,
573
00:39:12 --> 00:39:15
and v2 is zero,
one.
574
00:39:15 --> 00:39:22
And again, I'll use the same
basis for the outputs.
575
00:39:22 --> 00:39:26
Then I have to do this -- I can
find a matrix,
576
00:39:26 --> 00:39:30
it will be the matrix that we
would always think of,
577
00:39:30 --> 00:39:33
it would be the projection
matrix.
578
00:39:33 --> 00:39:37
It will be, actually,
it's the matrix that we learned
579
00:39:37 --> 00:39:41
about in chapter four,
it's what I call the matrix --
580
00:39:41 --> 00:39:45.52
do you remember,
P was A, A transpose over A
581
00:39:45.52 --> 00:39:47
transpose A?
582
00:39:47 --> 00:39:51
And I think,
in this example,
583
00:39:51 --> 00:39:55
it will come out,
one-half, one-half,
584
00:39:55 --> 00:39:57.76
one-half, one-half.
585
00:39:57.76 --> 00:40:04
I believe that's the matrix
that comes from our formula.
586
00:40:04 --> 00:40:10
And that's the matrix that will
do the job.
587
00:40:10 --> 00:40:17
If I give you this input,
one, zero, what's the output?
588
00:40:17 --> 00:40:21
The output is one-half,
one-half.
589
00:40:21 --> 00:40:26
And that should be the right
projection.
590
00:40:26 --> 00:40:33.33
And if I give you the input
zero, one, the output is,
591
00:40:33.33 --> 00:40:40
again, one-half,
one-half, again the projection.
592
00:40:40 --> 00:40:44
So that's the matrix,
but not diagonal of course,
593
00:40:44 --> 00:40:49
because we didn't choose a
great basis, we just chose the
594
00:40:49 --> 00:40:50
handiest basis.
595
00:40:50 --> 00:40:54
Well, so the course has
practically been about the
596
00:40:54 --> 00:40:57
handiest basis,
and just dealing with the
597
00:40:57 --> 00:41:00
matrix that we got.
598
00:41:00 --> 00:41:03
And it's not that bad a matrix,
it's symmetric,
599
00:41:03 --> 00:41:08
and it has this P squared equal
P property, all those things are
600
00:41:08 --> 00:41:09
good.
601
00:41:09 --> 00:41:13
But in the best basis,
it's easy to see that P squared
602
00:41:13 --> 00:41:17
equals P, and it's symmetric,
and it's diagonal.
603
00:41:17 --> 00:41:21
So that's the idea then,
is, do you see now how I'm
604
00:41:21 --> 00:41:25
associating a matrix to the
transformation?
605
00:41:25 --> 00:41:31
I'd better write the rule down,
I'd better write the rule down.
606
00:41:31 --> 00:41:34
The rule to find the matrix A.
607
00:41:34 --> 00:41:36
All right, first column.
608
00:41:36 --> 00:41:40
So, a rule to find A,
we're given the bases.
609
00:41:40 --> 00:41:45
Of course, we don't -- because
there's no way we could
610
00:41:45 --> 00:41:51
construct the matrix until we're
told what the bases are.
611
00:41:51 --> 00:41:58.34
So we're given the input basis,
and the output basis,
612
00:41:58.34 --> 00:42:00
v1 to vn, w1 to wm.
613
00:42:00 --> 00:42:02
Those are given.
614
00:42:02 --> 00:42:10
Now, in the first column of A,
how do I find that column?
615
00:42:10 --> 00:42:15
The first column of the matrix.
616
00:42:15 --> 00:42:21
So that should tell me what
happens to the first basis
617
00:42:21 --> 00:42:22
vector.
618
00:42:22 --> 00:42:27
So the rule is,
apply the linear transformation
619
00:42:27 --> 00:42:28
to v1.
620
00:42:28 --> 00:42:31
To the first basis vector.
621
00:42:31 --> 00:42:37
And then, I'll write it -- so
that's the output,
622
00:42:37 --> 00:42:38
right?
623
00:42:38 --> 00:42:43.46
The input is v1,
what's the output?
624
00:42:43.46 --> 00:42:48.55
The output is in the output
space, it's some combination of
625
00:42:48.55 --> 00:42:53
these guys, and it's that
combination that goes into the
626
00:42:53 --> 00:42:57
first column -- so,
let me -- I'll put this word --
627
00:42:57 --> 00:43:00
right, I'll say it in words
again.
628
00:43:00 --> 00:43:02.77
How to find this matrix.
629
00:43:02.77 --> 00:43:06
Take the first basis vector.
630
00:43:06 --> 00:43:11
Apply the transformation,
then it's in the output space,
631
00:43:11 --> 00:43:16
T of v1, so it's some
combination of these outputs,
632
00:43:16 --> 00:43:17
this output basis.
633
00:43:17 --> 00:43:21
So that combination,
the coefficients in that
634
00:43:21 --> 00:43:26
combination will be the first
column -- so a1,
635
00:43:26 --> 00:43:28
a row 2, column 1,
w2, am1, wm.
636
00:43:28 --> 00:43:35
There are the numbers in the
first column of the matrix.
637
00:43:35 --> 00:43:41
Let me make the point by doing
the second column.
638
00:43:41 --> 00:43:44
Second column of A.
639
00:43:44 --> 00:43:47
What's the idea,
now?
640
00:43:47 --> 00:43:55
I take the second basis vector,
I apply the transformation to
641
00:43:55 --> 00:44:03
it, that's in -- now I get an
output, so it's some combination
642
00:44:03 --> 00:44:11
in the output basis --
and that combination is the
643
00:44:11 --> 00:44:17
bunch of numbers that should go
in the second column of the
644
00:44:17 --> 00:44:18
matrix.
645
00:44:18 --> 00:44:18
OK.
646
00:44:18 --> 00:44:20
And so forth.
647
00:44:20 --> 00:44:25
So I get a matrix,
and the matrix I get does the
648
00:44:25 --> 00:44:26.43
right job.
649
00:44:26.43 --> 00:44:32
Now, the matrix constructed
that way, and following the
650
00:44:32 --> 00:44:37
rules of matrix multiplication.
651
00:44:37 --> 00:44:43
The result will be that if I
give you the input coordinates,
652
00:44:43 --> 00:44:49
and I multiply by the matrix,
so the outcome of all this is A
653
00:44:49 --> 00:44:56
times the input coordinates
correctly reproduces the output
654
00:44:56 --> 00:44:57
coordinates.
655
00:44:57 --> 00:44:59
Why is this right?
656
00:44:59 --> 00:45:02
Let me just check the first
column.
657
00:45:02 --> 00:45:09
Suppose the input coordinates
are one and all zeros.
658
00:45:09 --> 00:45:11
What does that mean?
659
00:45:11 --> 00:45:12
What's the input?
660
00:45:12 --> 00:45:16
If the input coordinates are
one and other -- and the rest
661
00:45:16 --> 00:45:19
zeros, then the input is v1,
right?
662
00:45:19 --> 00:45:23
That's the vector that has
coordinates one and all zeros.
663
00:45:23 --> 00:45:24
OK?
664
00:45:24 --> 00:45:27
When I multiply A by the one
and all zeros,
665
00:45:27 --> 00:45:32
I'll get the first column of A,
I'll get these numbers.
666
00:45:32 --> 00:45:35
And, sure enough,
those are the output
667
00:45:35 --> 00:45:37.19
coordinates for T of v1.
668
00:45:37.19 --> 00:45:40
So we made it right on the
first column,
669
00:45:40 --> 00:45:44
we made it right on the second
column, we made it right on all
670
00:45:44 --> 00:45:48
the basis vectors,
and then it has to be right on
671
00:45:48 --> 00:45:49.42
every vector.
672
00:45:49.42 --> 00:45:49
OK.
673
00:45:49 --> 00:45:53
So there is a picture of the
matrix for a linear
674
00:45:53 --> 00:45:55
transformation.
675
00:45:55 --> 00:46:01.97
Finally, let me give you
another -- a different linear
676
00:46:01.97 --> 00:46:03
transformation.
677
00:46:03 --> 00:46:10
The linear transformation that
takes the derivative.
678
00:46:10 --> 00:46:13
That's a linear transformation.
679
00:46:13 --> 00:46:21
Suppose the input space is all
combination c1 plus c2x plus c3
680
00:46:21 --> 00:46:23
x squared.
681
00:46:23 --> 00:46:27
So the basis is these simple
functions.
682
00:46:27 --> 00:46:30
Then what's the output?
683
00:46:30 --> 00:46:32
Is the derivative.
684
00:46:32 --> 00:46:38
The output is the derivative,
so the output is c2+2c3 x.
685
00:46:38 --> 00:46:44
And let's take as output basis,
the vectors one and x.
686
00:46:44 --> 00:46:49
So we're going from a
three-dimensional space of
687
00:46:49 --> 00:46:54
inputs to a two-dimensional
space of outputs by the
688
00:46:54 --> 00:46:57
derivative.
689
00:46:57 --> 00:47:07.56
And I don't know if you ever
thought that the derivative is
690
00:47:07.56 --> 00:47:08
linear.
691
00:47:08 --> 00:47:18
But if it weren't linear,
taking derivatives would take
692
00:47:18 --> 00:47:21
forever, right?
693
00:47:21 --> 00:47:25
We are able to compute
derivatives of functions exactly
694
00:47:25 --> 00:47:27
because we know it's a linear
transformation,
695
00:47:27 --> 00:47:31
so that if we learn the
derivatives of a few functions,
696
00:47:31 --> 00:47:34
like sine x and cos x and e to
the x, and another little short
697
00:47:34 --> 00:47:38
list, then we can take all their
combinations and we can do all
698
00:47:38 --> 00:47:40
the derivatives.
699
00:47:40 --> 00:47:42
OK, now what's the matrix?
700
00:47:42 --> 00:47:44
What's the matrix?
701
00:47:44 --> 00:47:49
So I want the matrix to
multiply these input vectors --
702
00:47:49 --> 00:47:53
input coordinates,
and give these output
703
00:47:53 --> 00:47:54
coordinates.
704
00:47:54 --> 00:47:58
So I just think,
OK, what's the matrix that does
705
00:47:58 --> 00:47:59
it?
706
00:47:59 --> 00:48:02.5
I can follow my rule of
construction,
707
00:48:02.5 --> 00:48:06
or I can see what the matrix
is.
708
00:48:06 --> 00:48:10
It should be a two by three
matrix, right?
709
00:48:10 --> 00:48:13
And the matrix -- so I'm just
figuring out,
710
00:48:13 --> 00:48:15
what do I want?
711
00:48:15 --> 00:48:17.75
No, I'll -- let me write it
here.
712
00:48:17.75 --> 00:48:20
What do I want from my matrix?
713
00:48:20 --> 00:48:22
What should that matrix do?
714
00:48:22 --> 00:48:26.14
Well, I want to get c2 in the
first output,
715
00:48:26.14 --> 00:48:29
so zero, one,
zero will do it.
716
00:48:29 --> 00:48:32
I want to get two c3,
so zero, zero,
717
00:48:32 --> 00:48:33.21
two will do it.
718
00:48:33.21 --> 00:48:37
That's the matrix for this
linear transformation with those
719
00:48:37 --> 00:48:39
bases and those coordinates.
720
00:48:39 --> 00:48:43
You see, it just clicks,
and the whole point is that the
721
00:48:43 --> 00:48:47
inverse matrix gives the inverse
to the linear transformation,
722
00:48:47 --> 00:48:51
that the product of two
matrices gives the right matrix
723
00:48:51 --> 00:48:54
for the product of two
transformations --
724
00:48:54 --> 00:49:03
matrix multiplication really
came from linear
725
00:49:03 --> 00:49:06
transformations.
726
00:49:06 --> 00:49:17
I'd better pick up on that
theme Monday after Thanksgiving.
727
00:49:17 --> 00:49:24
And I hope you have a great
holiday.
728
00:49:24 --> 00:49:30
I hope Indian summer keeps
going.
729
00:49:30 --> 00:49:33
OK, see you on Monday.