Linear Algebra and Its Applications · Problem 2.4.37

Vector Spaces — Gilbert Strang

Mathematics · four-fundamental-subspaces

Problem

True or false (with a reason or a counterexample)? (a)

A

and

A^T

have the same number of pivots. (b)

A

and

A^T

have the same left nullspace. (c) If the row space equals the column space then

A^T = A

. (d) If

A^T = -A

then the row space of

A

equals the column space.