Geometric Interpretation of Image What is the geometric Interpretation of the image of the following linear mapping?
               $f: \bf R^3 \to\bf R^3$  with $f(x,y,z)=f(x-y+z,-x+2y+z,y+2z)$
If you mean the entire image, it is a subspace of $\mathbb{R}^3$, which means it could be a point, a line through the origin, a plane through the origin, or the whole space. The answer is given by the rank of the matrix $\begin{bmatrix} 1 &amp; -1 &amp; 1 \\ -1 &amp; 2 &amp; 1 \\ 0 &amp; 1 &amp; 2 \end{bmatrix}$. If it is zero, the image is a point, if it is 1, the image is a line, etc. You can compute the rank by row reduction.
HINT:

Since z* = x* + y* , we have sphere mapped to a cylinder etc... (forgot its name)