- The rank of a matrix is the rank of the corresponding linear transformation.
- Full rank is a thing.
- Full rank if and only if it is injective or surjective.
- Recall row operations is left-multiplication by an invertible matrix.
- The range does not change when pre-composing by an isomorphism. What if not isomorphic?
- The nullspace does not change when post-composing by an isomorphism. What if not isomorphic?
- Therefore, rank does not change under column and row operations.
- The range is the column space.
- Read Theorem 3.6 on your own. After row and column operations, every matrix decomposes as I’s and O’s.
- Corollary: rank is invariant under transposes.
- Method of computing inverses. M(A|I)=(MA|MI)
- Cosets and the first isomorphism theorem.