Linear Transformations

Structure-Preserving Maps

A linear transformation is a map $T : V \to W$ between vector spaces that preserves the two operations: $$T(u + v) = T(u) + T(v), \qquad T(cv) = c\,T(v).$$ Transforming a sum equals summing the transforms, and scaling commutes with $T$. Rotations, reflections, scalings, projections, differentiation, and integration are all linear; bending a line into a curve, or adding a constant shift, is not.

An immediate consequence of linearity is that $T(\mathbf{0}) = \mathbf{0}$: a linear map fixes the origin (set $c = 0$ in the scaling rule). Any candidate map that moves the origin — like $T(v) = v + b$ with $b \neq \mathbf{0}$ — is therefore not linear.

The central structural fact: a linear transformation is completely determined by its action on a basis. Since every vector is a unique combination of basis vectors, and $T$ preserves combinations, knowing $T(e_1), T(e_2), \dots$ pins down $T$ everywhere. For maps between finite-dimensional spaces this gives the matrix representation: place the images $T(e_j)$ as the columns of a matrix $A$, and then $T(\mathbf{v}) = A\mathbf{v}$. A matrix is a linear transformation in coordinates.

Because transformations are matrices, composing transformations corresponds to multiplying matrices — and since order matters for transformations (rotate-then-reflect $\neq$ reflect-then-rotate), matrix multiplication is non-commutative, closing the loop with the matrix theory of Unit 3.

Common pitfall: assuming every map of vectors is linear, or forgetting that a linear map must fix the origin. Linearity is strict: both $T(u+v) = T(u)+T(v)$ and $T(cv) = cT(v)$ must hold, which forces $T(\mathbf{0}) = \mathbf{0}$. A map that adds a nonzero constant ($T(v) = v + b$) moves the origin and is not linear — even though it looks simple.