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

Let $A\in M_{m\times n}(F)$. Its row space $\operatorname{Row}(A)\subseteq F^n$ is the span of its rows, its column space $\operatorname{Col}(A)\subseteq F^m$ is the span of its columns, and its nullspace is $N(A):=\{x\in F^n:Ax=0\}.$

The row rank and column rank are $\operatorname{rrank}(A):=\dim_F\operatorname{Row}(A),\qquad \operatorname{crank}(A):=\dim_F\operatorname{Col}(A).$ These dimensions are defined because each space is spanned by a finite list. Starting with the empty list, inspect the spanning vectors in order and retain a vector exactly when it is not in the span of those already retained. Induction shows that the retained list is independent and has the same span as the original finite list, so it is a finite basis. The rank of $A$ is its row rank; Row rank equals column rank, and both equal the number of pivots proves that it also equals the column rank.