I have a linear algebra question I need help with.
Let $A$ be an $m\times m$ matrix with $\|A\|_2 < 1$ where $\|A\|_2$ is the $2$-norm of $A$. Show that $I - A$ is invertible where $I$ is the identity matrix.
I know that $\|Ax\|_2 \leq C\|x\|_2$ for some constant $C$ and a vector $x$. However I don't know the definition of $\|x\|_2$. I also don't see how this definition can help solve this problem.