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

already

there?

But don't be fled and

let your guard down,

it's going to get prey

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 lk

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

beer draw a

picture. What

do you say?

This should clear

things up a bit.

That’s a lile

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 lk 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 gd at

cking, so

most of the

time, yeah...

We, next time we

mt you won't

have to. I'm making

you lunch!

Don't woy

about it!

Tetsuo is sti

helping me

out and a!

You

don'twant

meto?

Oh...How

love ly.

I make a lot

ofthem for my

brother t, you

know—stamina

lunches.

On second

thought,

I’dlove for

you to...

...Make me

lunch.

Great!

No, that's not

it, it's just...

Uh...

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