Suppose A is a m x n matrix and dim(null(A)) = d. Determine dim(null(A transpose) and justify.
Thank You!
the null space of $\displaystyle A^T$ is equal to the orthogonal complement of the column space of $\displaystyle A.$ we know that the dimesnion of the column space of $\displaystyle A$ is equal to the rank of $\displaystyle A,$ which, by the rank-nullity
theorem, is equal to $\displaystyle n-d.$ thus the dimension of the orthogonal complement of the column space of $\displaystyle A$ is $\displaystyle m-n+d. \ \Box$