$A$ be a $m\times n$ matrix with row and column rank $r$, I need to know the dimension of the space of solution of the system of linear equation given by $AX=0$ by the approach of linear transformation.
I see this is a map from $\mathbb{R}^n\to \mathbb{R}^m $ given by $X\mapsto AX$, and hence I need to find the dimension kernel of this map.
am I right upto here?
Now how do I apply here $\dim \operatorname{Ker} T+\dim\operatorname{Im} T=\dim V$
Thank you.
Best Answer
We have $$\{\mathbf{x}\in\mathbb{R}^n|A\mathbf{x}=0\}=:\ker A$$ so the dimension you're looking for is $\dim\ker A$. As you have stated, we have the formula $$\dim \mathbb{R}^m = \dim\ker A+\dim A\mathbb{R}^n$$ Now notice that $\dim A\mathbb{R}^n=\operatorname{rank} A$, thus $$\dim\ker A = m - r$$