Prove that R(A) = R(B) iff N$(B^T)$ = N$(A^T)$ where R(A) denotes the range of the matrix A.

Question

This is a bonus question on my assignment and it bothered me quite a while.

I believe the range of a matrix is equivalent to the column space of the matric. That is to prove two matrices have the same row spaces iff their null spaces equal.

However, according to my understanding, the null space of a matrix is related to its column space(I doubted could it be the professor did some typographical error?). Ax means either the linear combination of the column vactors of A or the dot product of its row vectors. I tried to solve the question doing this to transfer between columns and rows(forgive my poor English), but it didn't work as I expected.

Then I run out of ideas

$A^T$ and $B^T$ are row equivalent, how is this indicate they have the same null space?

Answer 1

I've figured out an approach.
when $N(A^T) \subseteq N(B^T)$
let $a_j$ denotes the column vectors of A ; B = [$b_1$...$b_n$]
$\iff$ x$\in$$N(B^T)$ for x$\in$$N(A^T)$
$\iff$ $(B^T,a_j^T)$x = 0 and $B^T$x = 0
$\iff$ $N(B^T) = N(B^T,a_j^T)$
$\iff$ Rank($B^T$) = Rank$(B^T,a_j^T)$
$\iff$ Rank(B) = Rank$(B,a_j)$
$\iff$ $a_j$ = span($b_1$...$b_n$)
$\iff$ R(A) $\subseteq$ R(B)
$\iff$ $N(A^T) = N(B^T)$ gives R(A) = R(B)