Linear Transformations and Matrices

A linear transformation $T: V \to W$ is a mapping between vector spaces that preserves the operations of addition and scalar multiplication.

Properties of Linear Maps

$T$ is linear if for all $\mathbf{u}, \mathbf{v} \in V$ and $c \in F$:

1. $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$
2. $T(c\mathbf{v}) = cT(\mathbf{v})$

Every linear transformation between finite-dimensional vector spaces can be represented as a matrix.

Composition and Matrix Multiplication

If $T: U \to V$ and $S: V \to W$ are linear maps, their composition $S \circ T: U \to W$ is also linear. The matrix representing $S \circ T$ is the product of the matrices representing $S$ and $T$.

Change of Basis

The matrix representation of a linear map depends on the choice of bases for $V$ and $W$. If $[T]_B$ is the matrix in basis $B$, and $P$ is the transition matrix from basis $B'$ to $B$, then: $[T]_{B'} = P^{-1} [T]_B P$