Coordinates of a Change of Basis Let $\{x_1, x_2, x_3\}$ and $\{y_1, y_2, y_3\}$ be basis of of $\mathbb{R}^3$ with a basis transformation:
$$
y_1 = 2x_1 - x_2 - x_3 
$$
$$
y_2 = -x_2
$$ 
$$
y_3 = 2x_2 + x_3
$$
What are all the vectors that have the same coordinates with respect to the two bases?

If I understand the question correctly
it can be interpreted as a question about eigenvector of eigenvalue $=1$ of the following matrix
$$A=\begin{bmatrix}
2 &-1&-1\\
0&-1&0\\
0&2&1
\end{bmatrix}$$
note that $y=Ax$
and the answer to the original question is $v\in Span\{(1,0,1)\}$, note $Av=v$, i.e. in $x$-basis and in $y$-basis $v$ has exactly the same representation.
