Computations with a given linear transformation

#linear_algebra

Let $T : R^{3} \to R^{3}$ be the linear transformation defined by $T ([\begin{matrix} x \\ y \\ z \end{matrix}]) = [\begin{matrix} x + y \\ 2 z - x \\ y + 2 z \end{matrix}]$ .

Find the matrix that represents $T$ with respect to the standard basis for $R^{3}$ .
Find a basis for the kernel of $T$ .
Determine the rank of $T$ .

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Let $T:{\bf R}^3\to {\bf R}^3$ be the linear transformation defined by $T\left(\begin{bmatrix} x \\ y \\ z\end{bmatrix}\right) = \begin{bmatrix}  x+y \\ 2z-x \\ y+2z\end{bmatrix}$.
\begin{enumerate}[label=\alph*)]
	\item Find the matrix that represents $T$ with respect to the standard basis for ${\bf R}^3$.
	\item Find a basis for the kernel of $T$.
	\item Determine the rank of $T$.
\end{enumerate}