Linear algebra studies linear maps and the spaces they act on. A linear map sends lines through the origin to lines through the origin and respects addition and scaling. Matrices are concrete recipes for those maps once bases are chosen; eigenvalues and eigenvectors reveal stretch-and-align behavior that coordinates can hide.
Vectors and bases
A vector is an arrow you can add and scale — or, more abstractly, an element of a vector space. A basis is a minimal set of vectors that can build every vector in the space by linear combination. Coordinates are the recipe coefficients relative to a basis; change the basis and the numbers change, even though the geometric vector is the same.
Matrices as machines
Multiplying by a matrix $A$ applies a linear transformation. Columns of $A$ are the images of the standard basis vectors. Solving $A\mathbf{x}=\mathbf{b}$ asks which inputs produce output $\mathbf{b}$. Row reduction is bookkeeping for that question; determinants measure oriented volume scaling (and vanish precisely when the map collapses dimension).
Eigenvalues: invariant directions
An eigenvector satisfies $A\mathbf{v}=\lambda\mathbf{v}$ with $\mathbf{v}\neq\mathbf{0}$: the map stretches or flips $\mathbf{v}$ but does not tip it off its line. The scalar $\lambda$ is the eigenvalue. For a careful plain-language tour, see Eigenvalues in plain language.