7-23
Rank
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.