Q:

For an $n \times n$ invertible matrix $E$ and $\vec{x}, x \in \mathbb{R}^n$, does $E \vec{x} = \vec{y},\vec{y} \in \mathbb{R}^n$?

I guess to clarify, I'm wondering if the matrix vector multiplication of an invertible matrix and any vector $\in \mathbb{R}^n$ can produce all the vectors $\in \mathbb{R}^n$ (does this also mean that $\{E\vec{x}\}$ spans $\mathbb{R}^n$?).

A:

Yes, if you want to produce the vector $\vec{y}$, you just let $\vec{x} = E^{-1}\vec{y}$. Then
$$
E\vec{x} = EE^{-1}\vec{y} = \vec{y}.
$$

