Linear Transformations

A precise treatment of linear maps, matrices, kernels, images, composition, and basis dependence.

A linear transformation moves an entire grid while preserving lines and the origin. A square source grid maps to a skewed image grid. Basis vectors and one eigen-direction are highlighted, while a signed area tile represents the determinant’s scale factor.

Precise definition

A map $T:V\to W$ is linear if $T(au+bv)=aT(u)+bT(v)$ for all vectors $u,v$ and scalars $a,b$. Equivalently, it preserves vector addition and scalar multiplication. Linearity forces $T(0)=0$.

Notation and mathematical language

After bases are chosen, the matrix $[T]_{C\leftarrow B}$ has column $j$ equal to $[T(b_j)]_C$. Then $[T(v)]_C=[T]_{C\leftarrow B}[v]_B$. Composition corresponds to matrix multiplication in compatible bases, with the rightmost map applied first.

Conceptual picture

A linear map is determined completely by its values on a basis because every vector is a unique linear combination. Its kernel measures collapsed directions and its range measures reachable outputs.

Fully worked example

Interpretation and application

Linear transformations model rotations, projections, differentiation, encoding, and local approximations. The matrix is coordinate-dependent, while injectivity, rank, and eigenstructure represent the underlying map.