Linear transformation of a subspace How can I calculate the image of a linear transformation of a subspace?
Example:
Given a subspace $A$ defined by $x + y + 2z=0$, and a linear transformation defined by the matrix 
$$M=
\left(
\begin{matrix}
1 & 2 & -1\\
0 & 2 & 3\\
1 & -1 & 1\\
\end{matrix}\right)
$$
What is the image of the transformation of A by M?

x+ y+ 2x is more commonly written as 3x+ y so I am going to assume you mean x+ y+ 2z= 0.  From that equation, x= -y- 2z so any vector satisfying that can be written <-y- 2z, y, z>.  Applying M to that, 
$\begin{pmatrix}1 & 2 & -1 \\ 0 & 2 & 3 \\ 1 & -1 & 1 \end{pmatrix}\begin{pmatrix}-y- 2z \\ y \\ z \end{pmatrix}= \begin{pmatrix}-y- 2z+ 2y-z\\ 2y+ 3z \\ -y- 2z- y+ z\end{pmatrix}= \begin{pmatrix} y- 3z \\ 2y+ 3z \\ -2y- z\end{pmatrix}$ $= y\begin{pmatrix}1 \\2 \\-2 \end{pmatrix}+ z\begin{pmatrix}-3 \\ 3 \\-1 \end{pmatrix}$
