[Math] Prove that the dimension of row space equals to the dimension of column space of an $n\times n$ matrix

Question

Knowing that the row space of $A\in \mathbb{R}^{n\times n}$ equals $N(A)^\perp$ prove that the dimension of column space of a matrix equals its row space dimension.

So I'm trying to apply properties of dimensions in this proof like:

$\dim (U^{\perp}) = \dim V – \dim U $

But I don't know any dimension (except V which might be $n \times n$) Can someone give a hint in how to proceed to start this proof?

Thanks!

Answer 1

By the definition of $N(A)^\perp$, one has $$ \dim (N(A)^\perp)+\dim(N(A))=n. $$ By the rank–nullity theorem, $$ \dim (N(A))+\dim R(A)=n. $$ Now, by cancellation you have the proof.