Interpreting the meaning of the RREF of the column space of a matrix So honestly I genuinely cannot find answers for this like I've tried but I genuinely just cannot. So when I have a system of equations I can represent it with an augmented matrix right? And I can solve it by putting the matrix into reduced row echelon form right? Now all that makes sense to me but now I have a new question when I get a matrix $A$ just like say some question gave me a matrix $A$ and it wants me to find the column space of $A$ then that's the span of the column vectors but now I have two questions, 1. How would you all interpret the matrix in this case? Like I shouldn't interpret it as just a system of equations right since I'm looking for a column space I should interpret it as an array of column vectors correct?
Now the second question, since I'm interpreting it as an array of column vectors, what is putting this matrix into reduced row echelon form doing to it? It's not a system of equations where row operations are defined to just be normal stuff like subtracting equations, swapping them, multiplying both sides by a number, I'm interpreting it as an array of column vectors and when I'm putting it into row echelon form I'm basically just doing row operations on these column vectors? These vectors are totally getting changed and becoming totally different vectors now so I really just don't understand how to interpret the matrix? What am I really doing here when I'm doing all these row operations to this array of column vectors?
 A: Recall that for finding the row space of a matrix $A$, you just compute the Reduced Row Echelon Form (RREF) of $A$ and then take the non-zero rows of $\text{rref}(A)$ as vectors in the row space of $A$.
When you compute the Column Space of $A$, this is really equivalent to computing the Row Space of $A^T$ (transpose of $A$). This is one reason why you're still allowed to do row operations on this matrix since you are viewing the matrix $A$ in terms of rows, not columns anymore. Once you compute $\text{rref}(A^T)$, from the non-zero rows of this matrix you obtain the vectors in the Row Space of $A^T$. Redoing the transpose operation will then directly give you the vectors in the Column Space of the given matrix $A$.
As for your first question, the Column Space of $A$ can be interpreted as the range (or image) of the corresponding matrix transformation. e.g. if $A$ is an $m\times n$ matrix, i.e. $A:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$, and $\mathbf{x}$ is an $n\times1$ column vector, then the column space of $A$ is: $$C(A) = \{\mathbf{b}| A\mathbf{x}=\mathbf{b} \},$$
where $\mathbf{b}$ is an $m\times1$ column vector. So, the column space is really the space of all outputs from a matrix transformation.
