### Linear Transformation

##### 1. Transformation

Transformations are rules that convert one data into another data. We may denote a transformation by

T: XY,

where T is the transformation (also called map, function), X is the source set (or domain), and Y is the target set (or codomain). Taking any element xX as input, the transformation produces an element y = T(x) ∈ Y as output. We also call y the image (or value) of x under the transformation T. We may also include more specific information about the transformation by writing T: XY, xy. A more rigorous theory of transformations can be found here.

Example The following are some transformations in everyday life.

Age: people → number

IDs: student → number

IDp: professor → number

Instructor: course → professor

Maker: product → manufacturer

Capital City: country → city

Population: city → number

Example In mathematics, transformations are often given by formulae.

Square: RR, xx2

Reflection in x-axis: R2R2, (x, y) → (x, -y)

Polar-Cartesian: R2R2, (r, θ) → (rcosθ, rsinθ)

Root-Polynomial: R2R2, (x1, x2) → (x1 + x2, x1x2)

T: R3 R2, (x1, x2, x3) → (x1 + 2x2 + 3x3, 4x1 + 5x2 + 6x3)

Here are two examples of mathematical transformations given by descriptions instead of formulae.

 Sign: R → R, positive number → 1 negative number → -1 0 → 0
 Dirichlet function: R → R, rational number → 1 irrational number → 0