Let $A$ be an $n \times n$ invertible matrix and $v_1,v_2,...,v_m$ an element of $\mathbb{R}^{n}$. If $\{v_1,v_2,...,v_m\}$ spans $\mathbb{R}^{n}$ then $\{Av_1, Av_2,...,Av_m\}$ also spans $\mathbb{R}^{n}$.
How to prove that if $A$ is not invertible, then $\{Av_1, Av_2,...,Av_m\}$ does not span $\mathbb{R}^{n}$?
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$\begingroup$Suppose that $A$ is not invertible. Then, I claim $A$ is not surjective, by contradiction.
Suppose that $A$ were surjective. By the rank nullity theorem, $\dim \mathbb{R}^n = \dim \ker A + \dim \text{im} A$. Since $\dim \text{im} A =n$ by surjectivity, it follows that $\dim \ker A = 0\implies \ker A = \{ 0\}$, so $A$ is injective, hence invertible, which is a contradiction. Hence, we are done.
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