Q:

Linear transformation change of basis

Let $T:\mathbb{R}^3\to\mathbb{R}^3$ by $T(x,y,z)=(5x+3y-3z,-y,6x+3y-4z)$.
$1$. Let $w_1=(2,0,1)$ and $w_2=Tw_1$. Show that $W=span\{w_1,w_2\}$ is a $2$-dimensional subspace of $\mathbb{R}^3$.
$2$. Show that $T(W)=W$.
$3$. Find some vector $v$ such that $w_1,w_2,v$ is a basis for $\mathbb{R}^3$.
$4$. Find the matrix representation of $T$ with respect to the basis $w_1,w_2,v$.
For part $1$, $T=\begin{bmatrix}5&3&-3\\0&-1&0\\6&3&-4\end{bmatrix}$, so $w_2=Tw_1=\begin{bmatrix}5&3&-3\\0&-1&0\\6&3&-4\end{bmatrix}\begin{bmatrix}2\\0\\1\end{bmatrix}=\begin{bmatrix}7\\0\\8\end{bmatrix}$, so $W=span(\begin{bmatrix}2\\0\\1\end{bmatrix},\begin{bmatrix}7\\0\\8\end{bmatrix})=span(\begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}0\\0\\1\end{bmatrix})$. Therefore, $W$ is a 2-dimensional subspace of $\mathbb{R}^3$.
For part $2$, I didn't really get it. What does $T(W)$ mean?
For part $3$, by adding a vector $v=\begin{bmatrix}0\\1\\0\end{bmatrix}$,we will form a basis for $\mathbb{R}^3$.
For part $4$, should I use the change of basis method for linear transformation? I had a hard on doing this. Can someone help me to check I had done and show me what to do for part $2$ and part $4$? Thank you.

A:

For part 2, $T(W)=\{Tw\;|\;w\in W\}$.
For part 4, yes, use the change of basis formula. Remember that the columns of a change-of-basis matrix are the images of the basis vectors.

