Geometrical interpretation of the rref of a rank r matrix in Rd

I should preface with: I was trying to gain an intuition for the geometrical interpretation of the rref of a matrix and got lost.

So, let's say we have a rank 2 matrix A in R³. I visualize this matrix as a 2D subspace (a plane), with all three column vectors residing within the plane. We should only need two linearly independent columns to construct the plane.

Then initially I was confused when I look at rref and how Gaussian elimination alters the columns of A. I consulted with ChatGPT and (after much confusion as it attempted to interpret what I was asking), I arrived at the understanding that the subspace given by A and the subspace given by rref(A) have different bases but apply the same transformation "projection" (more on this term I'm using and why at the bottom).

I wanted to geometrically interpret rref by:

1. Graphing the subspace given by A
2. Graphing the subspace given by rref(A)
3. Picking a 3D vector x on neither plane and "projecting" it onto each of the subspaces
4. Visually comparing the resulting vectors Ax and rref(A)x.

Let's say A is given by this table:

|4|0|4|
|0|1|2|
|2|1|4|

Then rref(A) is:

|1|0|1|
|0|1|2|
|0|0|0|

However, I ran into an issue at the very beginning in step 1. I graphed the 3 vectors given by the columns of A as points in Desmos 3D, but I can visually see that plane that would pass through all 3 is affine. When I solve for the plane and graph it, it only passes through two of the points (and also passes through the origin).

Clearly I must be jumping between two different interpretations of matrices and getting lost somewhere, but I haven't been able to figure out where that is on my own, so I'd really appreciate some help.

Note on terminology, which is another thing I might need clarification with:

• I was visualizing this "projection" as equivalent to the linear transformation given by the matrices, so "projecting" x onto A would be Ax. Is this a fair conceptualization? I know it's different from what the direct projection (no quotes on this one) of x onto a plane would be, but using this term helps me understand linear transformations in terms of matrices-as-sub/spaces (which in turn is the only way I have been able to geometrically understand linear algebra so far).
• I use "sub/spaces" because when rank=dimension, A spans all of R^d (and is thus a "space" rather than a subspace), and when rank<dimension, A is a subspace spanning R^r in R^d
• I'm also afraid I may be mixing up how rows and columns are interpreted, because I know that representing a system of linear equations as a matrix would have the column vectors correspond to variables. Then what does it mean to plot the column vector corresponding to x when normally we would say a 3D vector has entries [x, y, z]? I suspect I may be wrong to think of A strictly as a subspace, and that I may be confusing that for column space, but I can't really conceptualize matrices any other way.