confusion about rank and nullity of matrix and rank-nullity theorem

Question

I can see that rank + nullity = number of columns of the matrix.

Does that mean the dimension of matrix is the number of columns?

Isn't dimension of a $m\times n$ matrix $mn$?

Thanks.

Answer 1

We don't typically talk about a matrix having a "dimension." An $m \times n$ matrix (with entries in $\mathbb{R}$) can be viewed as a linear map $L: \mathbb{R}^n \to \mathbb{R}^m$ given by left multiplication. That is, given such a matrix $A$ and an $n \times 1$ column vector $v$, the function $L$ maps $v \mapsto Av$, which is an $m \times 1$ column vector.

The dimension mentioned in the rank-nullity theorem is the dimension of the domain of this linear map (as a vector space), which is, as you point out, the number of columns of the matrix.

The set of all $m \times n$ matrices with entries in $\mathbb{R}$ does itself form a vector space, and its dimension is $mn$, as you also stated.