Having extended the euclidean spaces to general vector spaces, we may also extend linear transformations.
A transformation T: V → W 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).
id(u) = u: V → V
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: V → W
sending any vector in V to the zero vector in W is linear.
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 = 1 ≠ 0, where 0 is the constant zero function, and is the zero vector of the vector space F(R).
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: R2 → F(R)
of combining sine and cosine is linear. A more general statement about the linearity of C can be found here.
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 X → AXB 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).