Linear Algebra In A Jiffy

Every $n$-tuple in $\mathbb{F}^{n}$ has a first non-zero entry and a last non-zero entry. What do the positions of such entries in the elements of a subspace W of $\mathbb{F}^{n}$ reveal about W? It turns out, a great deal! This insight offers a beeline to the fundamental results on general bases and dimension (including dimensions of complements, rank of a transpose, Rank-Nullity Theorem, Full Rank Factorization, and existence/uniqueness of RREF), bypassing some of the traditional stumbling blocks and time sinks.