I am having a difficult time connecting different parts of a geometric interpretation of an underdetermined system of equations.
Given the matrix
$$A = \begin{bmatrix} 2 & 1 & 3\\ 1 & 2 & 3 \end{bmatrix}$$
I can interpret it as each column being a vector in 2-dimensional space in which case I can express any vector on that plane with the first and second column, since the third one is a linear combination of the first two.
I can also interpret the matrix as the equations of the planes
$$ \begin{aligned} 2x+y+3z&=0 \\ x+2y+3z&=0 \end{aligned}$$ and the solution for those equations being the line that is the intersection between the two planes. This gets me a line instead of a whole plane as above.
If I see it as the coefficient matrix for a linear transformation, then to use it I would have to provide a vector in 3D-space, and the output would land in plane in 3D-space.
If I am asked to provide the geometric interpretation of the above matrix, which of these would be correct? Is there some relation between them that I am not seeing? Does the question require more context to be answered?