Basic question on the infinitely many solutions of a linear system Ax=b, 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,

More generally, if $n$ is the nullity of the matrix corresponding to a homogeneous linear system, then the solution space is a $n$-dimensional subspace (it passes through the origin). If we have a corresponding consistent non-homogeneous system, the solution space is the affine space obtained from the aforementioned subspace by a translation through any particular solution.
