A vector space $V$ over a field $\mathbb F$ is a set with vector addition and scalar multiplication satisfying the usual axioms. A finite basis $\{e_1, \ldots, e_n\}$ gives a coordinate representation; the dimension $n$ is a basis-independent invariant.

A linear map $T: V \to W$ satisfies $T(\alpha u + \beta v) = \alpha T u + \beta T v$. After choosing bases, $T$ becomes a matrix.

Eigenvalues: $\lambda$ is an eigenvalue of $T$ if there exists nonzero $v$ with $Tv = \lambda v$. They are roots of the characteristic polynomial $\det(T - \lambda I) = 0$. The spectral theorem: a real symmetric (or complex Hermitian) matrix is diagonalizable by an orthonormal basis of eigenvectors with real eigenvalues.

Interactive: 2D linear transformation

Adjust the matrix entries — the grid is mapped by $T$. Eigenvectors keep their direction.