Finding a transformation with respect to a basis – does this question have no solution?

I have a problem that says:

Let
V={[xyz]| x+y+z=0}V =
\left\{
\begin{bmatrix}
x\\
y\\
z\\
\end{bmatrix}
\text{| $x +y+z = 0$}
\right\}

W={[xyz]| x+y=0}W =
\left\{
\begin{bmatrix}
x\\
y\\
z\\
\end{bmatrix}
\text{| $x +y = 0$}
\right\}

Find a basis for VV and WW, called α\alpha and β\beta respectively. Find β[T]α_\beta[T]_\alpha, where T:V→WT : V \rightarrow W is defined by:
T([xyz])=[x−yy+zy−x]T
\left(
\begin{bmatrix}
x\\
y\\
z\\
\end{bmatrix}
\right) =
\begin{bmatrix}
x – y\\
y + z\\
y – x\\
\end{bmatrix}

But is TT actually a transformation from VV to WW? Because (x−y)+(y+z)=x+z=−y(x-y) + (y + z) = x + z = -y, which is not necessarily 00. Am I just missing something here, or misunderstanding something?

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1

You are correct, the problem is iill posed. Note that v=(1,1,−2)T∈Vv=(1,1,-2)^T \in V but Tv=(0,−2,0)T∉WTv = (0,-2, 0)^T \notin W.
– copper.hat
Oct 21 at 3:07

I believe that youâ€™re completely correct. A general element of VV is (a,b,−b−a)(a,b,-b-a), and its image under TT is (a−b,−a,b−a)(a-b,-a,b-a), which will not be in WW unless b=0b=0.
– Lubin
Oct 21 at 3:10

Okay, thanks. Was trying to solve it and ran into issues, didn’t know if I just messed up with selecting α\alpha and β\beta somehow or if the problem was wrong.
– jgunter
Oct 21 at 3:19

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