Linear Transformations

Structure-Preserving Maps

A linear transformation is a map $T : V \to W$ between vector spaces that preserves vector addition and scalar multiplication: $$T(u + v) = T(u) + T(v), \quad T(cv) = c\,T(v).$$

Rotations, reflections, scalings, projections, differentiation, and integration are all linear; bending a line or adding a constant shift is not.

An immediate consequence is that fixes the origin ($T(\mathbf{0}) = \mathbf{0}$ by setting $c = 0$). Any map moving the origin is not linear.

Matrices and Composition

A linear transformation is completely determined by its action on a basis. Since every vector is a combination of basis vectors, knowing $T(e_1), T(e_2), \dots$ pins down $T$ everywhere.

For finite-dimensional spaces, we form the matrix representation by placing the images $T(e_j)$ as the columns of a matrix $A$, yielding $T(\mathbf{v}) = A\mathbf{v}$.

Because transformations are matrices, composing transformations corresponds to multiplying matrices. Since order matters (rotate-then-reflect $\neq$ reflect-then-rotate), matrix multiplication is non-commutative.

Common pitfall: Assuming every map is linear. Linearity is strict: both rules must hold simultaneously, forcing $T(\mathbf{0}) = \mathbf{0}$.