Row space, column space, nullspace, row rank, column rank and matrix rank

The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations

Linear combination of a finite list, and the span $\mathrm{span}(S)$ as the smallest linear subspace containing $S$

Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis

Linear independence: a finite list $v:n\to V$ is independent when ${\sum }_{i<n}{\lambda }_i{v}_i=0_V$ forces every ${\lambda }_i=0_F$, and a subset $S\subseteq V$ is independent when every injective finite list into $S$ is independent

Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis

The principle of mathematical induction

Matrix equation $Ax=b$, its solution set, consistency, homogeneous systems and the augmented matrix $[A\mid b]$