Interpret the solution of the system $Ax=b$ geometrically

Question

Let $A$ be a $2\times 3$ matrix and let $b\in \Bbb R^2$. Interpret the solution of the system $Ax=b$ geometrically, and discuss the possible cases in algebraic and geometric terms.

Algebraically, we have: $$\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}=\begin{bmatrix}a_{11}x_1+a_{12}x_2+a_{13}x_3\\a_{21}x_1+a_{22}x_2+a_{23}x_3\end{bmatrix}=\begin{bmatrix}b_1\\b_2\end{bmatrix}.$$ Algebraically, both of the induced equations are equations of a plane. Either they define the same equation, in which case an entire plane is sent to $b$, else if the planes intersect on a line, then that line is all that is sent to $b$, or else they don't intersect, and no points are sent to this point by $A$.

Do you agree with my argument here?

Answer 1

I don't know what you mean when you say that something is sent to $b$.

There are $3$ cases:

1. Both equations define the same plane. In that case, the solutions of the system are the points of that plane.
2. The equations define two parallel distinct planes. In that case, there are no solutions.
3. In the remaining cases, the equations define planes whose intersection is a line. Then the solutions of the system are the points of that line.

Answer 2

Taken separately and accounting for degenerate cases, the two equations define

• the whole space (augmented matrix of rank 0),

• a plane or

• the empty set (matrix of rank 0, augmented matrix of rank > 0).

Then there are nine possible combinations, among which the non trivial plane-plane case, which can yield

• a straight line (general case, matrix of rank 2),

• a plane (coinciding planes, matrix and augmented matrix of rank 1),

• the empty set (parallel planes, matrix of rank 1, augmented matrix of rank > 1).