1. ## Linear Algebra help

Suppose A is a m x n matrix and dim(null(A)) = d. Determine dim(null(A transpose) and justify.

Thank You!

2. Originally Posted by ishan
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 $A^T$ is equal to the orthogonal complement of the column space of $A.$ we know that the dimesnion of the column space of $A$ is equal to the rank of $A,$ which, by the rank-nullity

theorem, is equal to $n-d.$ thus the dimension of the orthogonal complement of the column space of $A$ is $m-n+d. \ \Box$