Question

if A is a mxn matrix of rank r, what are the dim N(A) and dim N(A^T)?

Answer 1

m-r and n-r

Answer 2

why? I understand why dim N(A) = m-r, but is it because A is transpose that it becomes n-r?

Answer 3

yes, for the transpose the rank is the same, but it is of course a n×m-matrix then.

Answer 4

thanks

Answer 5

actually isn't it dim N(A) = n-r and dim N(A^T) = m-r?

Answer 6

Oh yeah, mixed that up, sorry. Of course you have to subtract the rank from the domain-dimension and not from the target space dimension ^^