Linear transformations - Mathematics, Engineering Video Lecture - Engineering Mathematics

You now know what a
transformation is, so let's
introduce a special kind of
transformation called a linear
transformation.
It only makes sense that we have
something called a linear
transformation because we're
studying linear algebra.
We already had linear
combinations so we might as
well have a linear
transformation.
And a linear transformation,
by definition, is a
transformation-- which we
know is just a function.
We could say it's from the set
rn to rm -- It might be
obvious in the next video why
I'm being a little bit
particular about that, although
they are just
arbitrary letters -- where
the following two
things have to be true.
So something is a linear
transformation if and only if
the following thing is true.
Let's say that we have
two vectors.
Say vector a and let's
say vector b, are
both members of rn.
So they're both in our domain.
So then this is a linear
transformation if and only if
I take the transformation of
the sum of our two vectors.
If I add them up first, that's
equivalent to taking the
transformation of each of the
vectors and then summing them.
That's my first condition
for this to be a linear
transformation.
And the second one is, if I take
the transformation of any
scaled up version of a vector
-- so let me just multiply
vector a times some scalar
or some real number c .
If this is a linear
transformation then this
should be equal to c times
the transformation of a.
That seems pretty
straightforward.
Let's see if we can apply these
rules to figure out if
some actual transformations
are linear or not.
So let me define a
transformation.
Let's say that I have the
transformation T.
Part of my definition I'm going
to tell you, it maps
from r2 to r2.
So if you give it a
2-tuple, right?
Its domain is 2-tuple.
So you give it an x1 and an x2
let's say it maps to, so this
will be equal to, or it's
associated with x1 plus x2.
And then let's just say it's 3
times x1 is the second tuple.
Or we could have written this
more in vector form.
This is kind of our
tuple form.
We could have written it -- and
it's good to see all the
different notations that you
might encounter -- you could
write it a transformation of
some vector x, where the
vector looks like
this, x1, x2.
Let me put a bracket there.
It equals some new vector,
x1 plus x2.
And then the second component
of the new
vector would be 3x1.
That's a completely legitimate
way to express our
transformation.
And a third way, which I never
see, but to me it kind of
captures the essence of what
a transformation is.
It's just a mapping or
it's just a function.
We could say that the
transformation is a mapping
from any vector in r2 that looks
like this: x1, x2, to--
and I'll do this notation-- a
vector that looks like this.
x1 plus x2 and then 3x1.
All of these statements
are equivalent.
But our whole point of writing
this is to figure out whether
T is linearly independent.
Sorry, not linearly
independent.
Whether it's a linear
transformation.
I was so obsessed with linear
independence for so many
videos, it's hard to get it out
of my brain in this one.
Whether it's a linear
transformation.
So let's test our
two conditions.
I have them up here.
So let's take T of, let's say
I have two vectors a and b.
They're members of r2.
So let me write it.
A is equal to a1, a2, and
b is equal to b1, b2.
Sorry that's not a vector.
I have to make sure that
those are scalars.
These are the components
of a vector.
And b2.
So what is a1 plus b?
Sorry, what is vector
a plus vector b?
Brain's malfunctioning.
All right.
Well, you just add up
their components.
This is the definition
of vector addition.
So it's a1 plus b1.
Add up the first components.
And the second components is
just the sum of each of the
vector's second compnents.
a2 plus b2.
Nothing new here.
But what is the transformation
of this vector?
So the transformation of vector
a plus vector b, we
could write it like this.
That would be the same thing as
the transformation of this
vector, which is just a1
plus b1 and a2 plus b2.
Which we know it equals
a vector.
It equals this vector.
Or what we do is for the first
component here, we add up the
two components on this side.
So the first component here
is going to be these
two guys added up.
So it's a1 plus a2
plus b1 plus b2.
And then the second component
by our transformation or
function definition is just 3
times the first component in
our domain, I guess
you could say.
So it's 3 times the first one.
So it's going to be 3 times
this first guy.
So it's 3a1 plus 3b1.
Fair enough.
Now what is the transformation
individually of a and b?
So the transformation of a is
equal to the transformation of
a -- let me write it this
way -- is equal to the
transformation of a1
a2 in brackets.
That's another way of
writing vector a.
And what is that equal to?
That's our definition of our
transformation right up here,
so this is going to be equal to
the vector a1 plus a2 and
then 3 times a1.
It just comes straight out
of the definition.
I essentially just replaced
an x with a's.
By the same argument, what
is the transformation
of our vector b?
Well, it's just going to be the
same thing with the a's
replaced by the b's.
So the transformation of our
vector b is going to be -- b
is just b1 b2 -- so it's
going to be b1 plus b2.
And then the second component in
the transformation will be
3 times b1.
Now, what is the transformation
of vector a
plus the transformation
of vector b?
Well, it's this vector
plus that vector.
And what is that equal to?
Well, this is just pure vector
addition so we just add up
their components.
So it's a1 plus a2
plus b1 plus b2.
That's just that component
plus that component.
The second component is 3a1 and
we're going to add it to
that second component.
So it's 3a1 plus 3b1.
Now, we just showed you that if
I take the transformations
separately of each of the
vectors and then add them up,
I get the exact same thing as
if I took the vectors and
added them up first and then
took the transformation.
So we've met our
first criteria.
That the transformation of the
sum of the vectors is the same
thing as the sum of the
transformations.
Now let's see if this works
with a random scalar.
So we know what the
transformation of a looks like.
What does ca look like,
first of all?
I guess that's a good
place to start.
c times our vector a is going
to be equal to c times a1.
And then c times a2.
That's our definition of scalar
multiplication time's a vector.
So what's our transformation --
let me go to a new color.
What is our -- let me do a color
I haven't used in a long
time, white.
What is our transformation
of ca going to be?
Well, that's the same thing as
our transformation of ca1, ca2
which is equal to a new vector,
where the first term
-- let's go to our definition
-- is you sum the first and
second components.
And then the second term is 3
times the first component.
So our first term
you sum them.
So it's going to be
ca1 plus ca2.
And then our second term
is 3 times our first
term, so it's 3ca1.
Now, what is this equal to?
This is the same thing.
We can view it as factoring
out the c.
This the same thing as c times
the vector a1 plus a2.
And then the second
component is 3a1.
But this thing right here,
we already saw.
This is the same thing as
the transformation of a.
So just like that, you see that
the transformation of c
times our vector a, for any
vector a in r2 -- anything in
r2 can be represented this way
-- is the same thing as c
times the transformation of a.
So we've met our second
condition, that when you when
you -- well I just stated it, so
I don't have to restate it.
So we meet both conditions,
which tells us that this is a
linear transformation.
And you might be thinking,
OK, Sal, fair enough.
How do I know that all
transformations aren't linear
transformations?
Show me something
that won't work.
And here I'll do a very
simple example.
Let me define my
transformation.
Well, I'll do it from r2
to r2 just to kind
of compare the two.
I could have done it from r to r
if wanted a simpler example.
But I'm going to define
my transformation.
Let's say, my transformation
of the vector x1, x2.
Let's say it is equal to
x1 squared and then
0, just like that.
Let me see if this is a
linear transformation.
So the first question
is, what's my
transformation of a vector a?
So my transformation of a vector
a-- where a is just the
same a that I did before--
it would look like this.
It would look like a1 squared
and then a 0.
Now, what would be my
transformation if I
took c times a?
Well, this is the same thing as
c times a1 and c times a2.
And by our transformation
definition -- sorry, the
transformation of c times this
thing right here, because I'm
taking the transformation
on both sides.
And by our transformation
definition this will just be
equal to a new vector that
would be in our codomain,
where the first term is
just the first term
of our input squared.
So it's ca1 squared.
And the second term is 0.
What is this equal to?
Let me switch colors.
This is equal to c squared
a1 squared and
this is equal to 0.
Now, if we can assume that c
does not equal 0, this would
be equal to what?
Actually, it doesn't
even matter.
We don't even have to make
that assumption.
So this is the same thing.
This is equal to c squared times
the vector a1 squared 0.
Which is equal to what?
This expression right here
is a transformation of a.
So this is equal to c squared
times the transformation of a.
Let me do it in the
same color.
So what I've just showed
you is, if I take the
transformation of a vector being
multiplied by a scalar
quantity first, that that's
equal to -- for this T, for
this transformation that I've
defined right here -- c
squared times the transformation
of a.
And clearly this statement right
here, or this choice of
transformation, conflicts with
this requirement for a linear
transformation.
If I have a c here I should
see a c here.
But in our case, I have
a c here and I
have a c squared here.
So clearly this negates
that statement.
So this is not a linear
transformation.
And just to get a gut feel
if you're just looking at
something, whether it's going to
be a linear transformation
or not, if the transformation
just involves linear
combinations of the different
components of the inputs,
you're probably dealing with
a linear transformation.
If you start seeing things where
the components start
getting multiplied by each
other or you start seeing
squares or exponents, you're
probably not dealing with a
linear transformation.
And then there's some functions
that might be in a
bit of a grey area, but it
tends to be just linear
combinations are going to lead
to a linear transformation.
But hopefully that gives you
a good sense of things.
And this leads up to what I
think is one of the neatest
outcomes, in the next video.