# Linear (vector) Transformations - What is meant by all vectors?

The equation $A\vec x = \vec b$ is the same thing as saying we transform $\vec x$ to $\vec b$ by means of multiplication by matrix $A$.

Let $\vec x \in \Bbb R^4$.

Let $T: \Bbb R^4 \rightarrow \Bbb R^2 | T(x) = A\vec x. \\$

My question: is the vector $\vec x \in \Bbb R^4$ the set of all possible combinations $(x_1, x_2, x_3, x_4)$, or does $\vec x$ point to one specific vector, e.g., $\begin{bmatrix} 1\\ 1\\ 1\\ 1\\ \end{bmatrix}$?

Textbook says "[transformation] accounts to finding all vectors $\vec x \in \Bbb R^4$ that are transformed into $\vec b$ ."

If you are solving the equation $A\vec{x}=\vec{b}$, then you want to find all vectors $\vec{x}$ which when multiplied by $A$ give the vector $\vec{b}$.
If you are defining a linear transformation $T(\vec{x})=A\vec{x}$, you are describing how $T$ acts on any possible vector $\vec{x}$.