I just want to verify the geometry of solutions to $Ax=b$, for the case when we have infinitely many solutions:
If say for a $3\times 3$ matrix, after Gaussian Elimination, I have two pivot variables and one free variable – for simplicity, assume the free variable is the third coordinate $x_3$.
Then the infinitely many solutions form a line in $\mathbb{R}^3$.
Is there a similar geometric intuition for when I have only one pivot variable, and two free variables? Do the infinitely many solutions form a plane in $\mathbb{R}^3$? I am pretty sure it does, but just wanted to ask the question, in case I am mistaken. The solutions would be in the form of linear combinations of two linearly independent vectors, so I think these two vectors will span a plane.
Thanks,
Best Answer
If you have one pivot variable, your reduced (augmented) matrix is of the form
$$\left[\begin{array}{ccc|c}a&b&c&d\\0&0&0&0\\0&0&0&0\end{array}\right]$$
then the solution is of the form $ax+by+cz = d$, which is the equation of a plane.