Definition of linear transformation/homomorphism

Definition: A function (T: V o W) between vector spaces over the same field is a linear transformation (also called a homomorphism) if it satisfies:

1. (T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})) for all (\mathbf{u}, \mathbf{v} \in V)
2. (T(c\mathbf{v}) = cT(\mathbf{v})) for all (\mathbf{v} \in V) and (c \in \mathbb{F})

Equivalently, (T) preserves linear combinations:
[T(c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k) = c_1T(\mathbf{v}_1) + \cdots + c_kT(\mathbf{v}_k)]

The term homomorphism emphasizes that (T) is a structure-preserving map between algebraic structures (vector spaces). It preserves the operations of addition and scalar multiplication.

Example: The derivative operator (D: P_n o P_{n-1}) defined by (D(p) = p') is a linear transformation because ((af + bg)' = af' + bg').