Definition of linear transformation/homomorphism

**Definition:** A function \(T: V \to W\) 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 \(D: P_n \to P_{n-1}\) defined by \(D(p) = p'\) is a linear transformation because \((af + bg)' = af' + bg'\).Copy