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50 Chapter 2 The Fundamentals
I know it's late, but
I'd also like to talk
a bit about linear
transformations if
you're okay with it.
Linear
transformations?
Oh right, one
of the main
subjects.
We're
there?
But don't be fled and
let your guard down,
it's going to get prey
abstract from now on!
O-okay!
Fundamentals
Matrices
Linear
Transformations
Eigenvalues and
eigenvectors
BasicsPrepMain
We' go into more
detail later on.
No, we're just going
to have a quick lk
for now.
Linear Transformations
Functions 51
Behold...linear
transformations!
Gah...
Linear Transformations
Let x
i
and x
j
be two arbitrary elements of the set X, c be any real number, and
f be a function from X to Y. f is called a linear transformation from X to Y if it
satisfies the following two conditions:
u f(x
i
)+ f(x
j
) = f(x
i
+ x
j
)
v cf(x
i
) = f(cx
i
)
H...
so that
means...
I think we'd
beer draw a
picture. What
do you say?
This should clear
things up a bit.
That’s a lile
easier to
understand...
Condition u is
when the sum of
these two
equals this.
And condition v
is when the
product of
this and a
scalar
c
equals this.
52 Chapter 2 The Fundamentals
An example of a linear transformation
An example of a function that is not a linear transformation
Let's have a lk at a couple of examples.
The function f(x) = 2x is a linear transformation. This is because it satisfies
both
and
, as you can see in the table below.
Condition
Condition
f(x
i
) + f(x
j
) = 2x
i
+ 2x
j
f(x
i
+ x
j
) = 2(x
i
+ x
j
) = 2x
i
+ 2x
j
cf(x
i
) = c(2x
i
) = 2cx
i
f(cx
i
) = 2(cx
i
) = 2cx
i
The function f(x) = 2x − 1 is not a linear transformation. This is because it
satisfies neither
nor
, as you can see in the table below.
Condition
Condition
f(x
i
) + f(x
j
) = (2x
i
− 1) + (2x
j
− 1) = 2x
i
+ 2x
j
2
f(x
i
+ x
j
) = 2(x
i
+ x
j
) − 1 = 2x
i
+ 2x
j
− 1
cf(x
i
) = c(2x
i
− 1) = 2cx
i
c
f(cx
i
) = 2(cx
i
) − 1 = 2cx
i
− 1
...
Are you
a right?
Yeah,
don't
woy!
Thanks,
Reiji.
My
pleasure.
Phew
That's
a
for
to day.
Functions 53
Reiji?
Um...
Do you always
eat lunch in
the schl
cafeteria?
I live alone,
and I'm not
that gd at
cking, so
most of the
time, yeah...
We, next time we
mt you won't
have to. I'm making
you lunch!
Don't woy
Tetsuo is sti
helping me
out and a!
You
don'twant
meto?
Oh...How
love ly.
I make a lot
ofthem for my
brother t, you
know—stamina
lunches.
On second
thought,
I’dlove for
you to...
...Make me
lunch.
Great!
No, that's not
it, it's just...
Uh...

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