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.