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M02.7 · Inverses, rank and the four subspaces

A free, illustrated module of 4 lessons and one interactive in M02 Linear Algebra I: Vectors and Matrices: The inverse: undoing a map; Column space: what a map can reach; Null space: what a map destroys; Four subspaces and the counting rule.

M02.7 L1 · Core 4 lessons · about 100 min Live · in review

Part of M02 Linear Algebra I: Vectors and Matrices in the Mathematics track.

Some maps can be run backwards and some cannot, and the difference is whether anything was lost on the way. These four lessons make that precise. They build the inverse matrix, then name what a matrix can reach (its column space), what it destroys (its null space), and how much of each there is (its rank). The module ends with the four subspaces of a matrix and the counting rule that ties them together: every input direction is either kept or lost.

What you need first. Matrices as linear maps (M02.4), matrix multiplication and the identity matrix (M02.5), and elimination with pivots and free variables (M02.6). Span, independence and basis (M02.2) are used from the second lesson on, and the dot product (M02.3) in the last.

Lessons

1. The inverse: undoing a map M02.7.1 25 min

   A linear map can be undone exactly when it loses nothing: no two inputs share an output. Meet the inverse matrix, derive the formula for a 2-by-2 inverse step by step, prove that the inverse of a product reverses the order, and see why a map that squashes can never be run backwards.

2. Column space: what a map can reach M02.7.2 25 min

   The outputs of a matrix are exactly the combinations of its columns. That set is the column space. Learn to picture it as a line, a plane or the whole space, decide whether Ax = b can be solved by asking whether b lies in it, and find its dimension, the rank, by counting pivots.

3. Null space: what a map destroys M02.7.3 25 min

   The null space of a matrix is the set of inputs it sends to zero. See why it is always a line, a plane or another subspace through the origin, compute it from the reduced form with free variables, and learn why every solution of Ax = b is one particular solution plus a vector of the null space.

4. Four subspaces and the counting rule M02.7.4 25 min

   Every matrix has four subspaces: the row space and null space among its inputs, the column space and left null space among its outputs. Derive the counting rule, rank plus nullity equals the number of columns, from pivots and free variables, and prove that the row space is perpendicular to the null space.

The interactive

M02.7-X1 · Subspace viewer. Change a 2 by 2 or 2 by 3 matrix and watch its null space among the inputs and its column space among the outputs. The rank and the dimension count update as you move, and one button slides an input along the null space while its output stays still. It lives in Four subspaces and the counting rule.