Unit 7: Inverse matrices, column space, and null space
About this unit
Solving Ax = v asks which input A sends to v. If det A ≠ 0, the inverse undoes A and there is exactly one answer; if det A = 0, the column space says which v can be reached and the null space says how many inputs reach it.
By the end, you can: Read a system of equations as Ax = v, solve it with the inverse when det A ≠ 0, and use the column space and null space when det A = 0.
Modules
5 modulesLearn
- Undoing a matrixArticle · A system of equations is Ax = v: find the input A sends to v. When det A ≠ 0 the inverse A⁻¹ undoes A, so x = A⁻¹v.Not started
- When nothing undoes itArticle · With det A = 0 the outputs fill only the column space, whose dimension is the rank; the null space is what goes to zero, and the two decide how many solutions there are.Not started
- Flatten the planeExperiment · Drag a column of an invertible matrix until det A = 0 and watch every output fall onto one line.Not started
- Three questions about Ax = vReview · Inverse, column space and null space, and what each says about solutions.Not started