Is this a valid linear transformation? (1) Which of the following is a linear transformation?
$ \quad $ (a) $ T ( \begin{pmatrix} x \\ y \end{pmatrix} )= \begin{pmatrix}  0 \\ x \\y \end{pmatrix}$, $ \quad $ (b) $ T ( \begin{pmatrix} x \\ y \\ z \end{pmatrix} ) = \begin{pmatrix}  y \\ x \end{pmatrix}$ $ \quad $ (c) $ T (\begin{pmatrix} x \\ y \end{pmatrix} )= \begin{pmatrix}  x^2 \\ y^2 \\ 0 \end{pmatrix}$
$ \quad $ (d) $ T (\begin{pmatrix} x \\ y \end{pmatrix}) = \begin{pmatrix}  1 \\ x \\y \end{pmatrix}$ $ \quad $ (e) $ T ( \begin{pmatrix}\ x \\ y  \\ z \end{pmatrix} ) = x \begin{pmatrix}  1 \\ 1 \\ z \end{pmatrix}$
I am new to this topic, am I correct in saying that only (a) is a linear transformation?
(2) $ T (\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}) = \begin{pmatrix}  2 \\ 2 \\2 \end{pmatrix}$ , $ \quad $ $ T (\begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}) = \begin{pmatrix}  1 \\ -1 \\ -1 \end{pmatrix}$, $ \quad $ $ T (\begin{pmatrix} 0 \\ 1 \\ -2 \end{pmatrix}) = \begin{pmatrix}  0 \\ 0 \\ 0 \end{pmatrix}$ $ \quad $
$ \quad $$ M = \begin{pmatrix}1 & 1 & 0 \\ 2 & 1 & 1 \\ 3 & 2 & -2  \end{pmatrix}. $
$ \quad $ Then,  $ T (\begin{pmatrix} x \\ y \\ z \end{pmatrix}) = \begin{pmatrix}  2 &  1 & 0 \\ 2 & -1 & 0 \\ 2 & -1 & 0 \end{pmatrix} M^{-1}  \begin{pmatrix} x \\ y \\ z \end{pmatrix} $  ?

$(a)$ and $(b)$ are.  The others do not possess the properties that $T(cv)=cT(v)$ and $T(v+w)=T(v)+T(w)$.
