Finding Bases

Linear Algebra (Matrices) · Axiom Academy

Row-reduce, keep the pivot columns, and read off a basis and the dimension. Find a basis for the subspace of , and state its dimension, where Nice work — you turned a spanning set into a genuine basis by stripping out the redundancy. Vectors as columns, then row-reduce: a column is a pivot column exactly when its vector is independent of the earlier ones. Pivot columns give the basis: the original vectors sitting over the pivot columns, , form a basis. Free columns are redundant: has no pivot, so it is a combination of the others ( ) and is safe to drop. Dimension = number of pivots = rank: three pivots means , so here W is all of . The same recipe works for any span or column space: row-reduce, take the pivot-column vectors, count them.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.