### Linear Transformation

##### 1. Definition

Having extended the euclidean spaces to general vector spaces, we may also extend linear transformations.

A transformation T: VW between two vector spaces is a linear transformation if

T(u + v) = T(u) + T(v), T(cu) = cT(u).

Since linear transformations between euclidean spaces are equivalent to matrices, the extension can be thought of as a generalization of matrices. Moreover, by taking c = 0 in T(cu) = cT(u), we have

T(0) = 0.

By combining the two properties, T also preserves linear combinations

T(c1u1 + c2u2 + ... + ckuk) = c1T(u1) + c2T(u2) + ... + ckT(uk).

Example For any vector space V, the identity transformation

id(u) = u: VV

is linear. More generally, any scaling map, such as u → 2u is also linear.

For any vector spaces V and W, the zero transformation

O(u) = 0: VW

sending any vector in V to the zero vector in W is linear.

Example The transformation

D(f) = f': C1(R) → C(R)

of taking derivatives is linear. The transformation

M(f) = (t2+1)f: F(R) → F(R)

of multiplying the function t2+1 is also linear.

The exponential transformation

P(f) = ef: F(R) → F(R)

is not linear because P(0) = e0 = 10, where 0 is the constant zero function, and is the zero vector of the vector space F(R).

Example The evaluation

E(f) = f(2): F(R) → R

at t = 2 is linear. The multiple evaluations

E(f) = (f(0), f(1), f(2)): F(R) → R3

at t = 0, 1, 2 is also linear. The transformation

C(a, b) = a cost + b sint: R2F(R)

of combining sine and cosine is linear. A more general statement about the linearity of C can be found here.

Example The transpose transformation

T(X) = XT: M(m, n) → M(n, m)

is linear. The transformation

M(X) = AX: M(k, n) → M(m, n)

of multiplying a fixed m by k matrix A on the left is also linear. More generally, for fixed A and B, the transformation XAXB is also linear. The transformation

S(X) = X2: M(n, n) → M(n, n)

of taking squares is not linear because S(2X) = 4X2 ≠ 2X2 = 2S(X).