Linear Maps and Matrices

Entry conditions

You should know the definition of a vector space and be comfortable with bases and dimension.

Definitions

A linear map (or linear transformation) $T : V \to W$ between vector spaces over the same field preserves addition and scalar multiplication: $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ and $T(a\mathbf{v}) = aT(\mathbf{v})$.

Given bases for $V$ and $W$, every linear map corresponds to a matrix — a rectangular array of scalars. Composing linear maps corresponds to multiplying their matrices.

The determinant of a square matrix measures the factor by which the map scales volumes. A matrix is invertible if and only if its determinant is nonzero.

An eigenvalue of a linear map $T$ is a scalar $\lambda$ such that $T(\mathbf{v}) = \lambda \mathbf{v}$ for some nonzero vector $\mathbf{v}$ (the eigenvector). Eigenvalues reveal the directions along which a linear map acts by pure scaling.

Vocabulary (plain language)

• Linear map: a function between vector spaces that respects addition and scaling.
• Matrix: a table of numbers representing a linear map relative to chosen bases.
• Determinant: a number encoding whether a matrix is invertible and how it scales volume.
• Eigenvalue: a scalar $\lambda$ for which $T(\mathbf{v}) = \lambda\mathbf{v}$ has a nonzero solution.

Intuition

A linear map is completely determined by where it sends the basis vectors. The matrix is just a bookkeeping device that records this information. Matrix multiplication encodes composition of linear maps.

In category theory, the category of vector spaces over $F$ has vector spaces as objects and linear maps as morphisms. Composition is composition of linear maps (matrix multiplication).

Common mistakes

• Confusing a linear map with its matrix. The map is basis-independent; the matrix depends on the choice of basis.
• Assuming every matrix has eigenvalues over $\mathbb{R}$. Some eigenvalues may be complex.