Linear Mappings

A linear mapping (or linear transformation) between vector spaces $V$ and $W$ is a function $T : V \to W$ that respects both vector operations: for all $\mathbf{u}, \mathbf{v} \in V$ and scalars $\lambda$, $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ and $T(\lambda \mathbf{v}) = \lambda T(\mathbf{v})$. Geometrically, linear maps send lines through the origin to lines through the origin; parallel lines remain parallel; the origin stays fixed.

The fundamental theorem of linear maps is this: every linear map between finite-dimensional vector spaces, once a basis is fixed, is multiplication by a matrix. Conversely, every matrix defines a linear map. This turns abstract geometric transformations into concrete computations — a deeply practical fact for any ML implementation.

Two subspaces are attached to every linear map. The kernel (or null space) $\ker(T) = \{\mathbf{v} \in V : T(\mathbf{v}) = \mathbf{0}\}$ measures information lost; a larger kernel means more inputs collapse to the same output. The image (or range) $\text{im}(T) = \{T(\mathbf{v}) : \mathbf{v} \in V\}$ is the set of possible outputs. Rank-nullity lives here: $\dim(V) = \dim(\ker T) + \dim(\text{im} T)$.

A linear map is injective (one-to-one) iff its kernel is trivial (just $\mathbf{0}$), surjective (onto) iff its image fills all of $W$, and bijective (an isomorphism) iff both. Two finite-dimensional vector spaces are isomorphic iff they have the same dimension — a stunning result that says, up to relabeling, there is only one real vector space of each dimension.

Worked example — shear on a square's corners: Apply the shear $S = \begin{pmatrix}1 & 1\\0 & 1\end{pmatrix}$ to the unit square's corners $(0,0), (1,0), (1,1), (0,1)$. Results: $(0,0), (1,0), (2,1), (1,1)$. The bottom edge stays fixed; the top edge slides right by 1 — parallelogram from a square. Note lines remain lines and parallelism is preserved.

Changing the basis of the domain or codomain transforms the matrix of a linear map by similarity: $A' = P^{-1} A P$ for a change-of-basis matrix $P$. Similar matrices represent *the same linear map* in different coordinate systems and share invariants like rank, determinant, trace, and eigenvalues. This is the theoretical foundation for diagonalization — the secret sauce behind PCA, eigenfaces, and spectral methods.