Linear map

• categories: Linear algebra, Definition

A linear map (or linear transformation) is a function $T:V\to W$ between two vector spaces $V$ and $W$ over the same field (typically $\mathbb{R}$ or $\mathbb{C}$) that preserves vector addition and scalar multiplication.

Definition

A function $T:V\to W$ is linear if, for all $x,y\in V$ and scalars $\alpha \in \mathbb{R}$ (or $\mathbb{C}$):

1. Additivity: $T(x+y)=T(x)+T(y)$,
2. Homogeneity: $T(\alpha x)=\alpha T(x)$.

Key Properties

• Zero Map: $T(0)=0$, since $T(0)=T(0\cdot x)=0\cdot T(x)=0$.
• Linearity of Composition: If $S:W\to Z$ is also linear, then $S\circ T:V\to Z$ is linear.
• Matrix Representation: If $V$ and $W$ are finite-dimensional, $T$ can be represented as a Matrix that describes its action in terms of basis vectors.

Examples

1. Scaling: $T(x)=\alpha x$ for a fixed scalar $\alpha$ is linear.
2. Differentiation: $D(f)=f^{\prime }$ is a linear map from the space of differentiable functions to itself.