Understanding the column space [closed]

$A\vec{x}$ is a linear combination of the columns of $A$, with the coefficients coming from $\vec{x}$ (this follows from the definition of matrix multiplication). In the example above, the multiplication $$\begin{bmatrix} 2 & 5 & 3\\ 4 & 0 & 8\\ 1 & 3 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix}$$ is the same thing as $$x \begin{bmatrix} 2\\4\\1 \end{bmatrix} + y \begin{bmatrix} 5\\0\\3 \end{bmatrix} + z \begin{bmatrix} 3\\8\\0 \end{bmatrix}.$$

If we look at it this way, we see how the range/image is equal to the span of the columns (i.e. the column space): $$ \text{range of $A$} = \{ A\vec{x}: \vec{x} \in \mathbb{R}^{3} \} = \{ \begin{bmatrix} 2 & 5 & 3 \\ 4 & 0 & 8\\ 1 & 3 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix}: x, y, z \in \mathbb{R} \} = \{ x \begin{bmatrix} 2\\4\\1 \end{bmatrix} + y \begin{bmatrix} 5\\0\\3 \end{bmatrix} + z \begin{bmatrix} 3\\8\\0 \end{bmatrix}: x, y, z \in \mathbb{R} \} = \text{column space of $A$} $$

You can see how this generalizes to any matrix and any vector.

Edit: to answer your "P4": with the view of matrix-vector multiplication as a linear combination of columns, you can see that $A\hat{i}$ is just the first column of $A$; $A\hat{j}$ the second column; and $A\hat{k}$ the third column. So, yes, the column space is $\{ \vec{V} \mid \vec{V} = c_1\vec{I} + c_2\vec{J} + c_3\vec{K} \}$.

In other words, the columns of $A$ are the transformed standard basis vectors, and the column space is the set of all linear combinations of these transformed vectors. So, in $\mathbb{R}^3$ for example, you can determine what a matrix does just by computing what it does to $\hat{i}, \hat{j}$, and $\hat{k}$. Then for any vector $\vec{x} = \begin{bmatrix}x_1 \\ x_2 \\ x_3 \end{bmatrix} = x_1\hat{i} + x_2\hat{j} + x_3\hat{k}$ in $\mathbb{R}^3$, $A\vec{x}$ is just $x_1 A\hat{i} + x_2A\hat{j} + x_3A\hat{k}$. This can be seen to be true by linearity, or by the view as described earlier of matrix-vector multiplication as a linear combination of the columns.