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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?

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All you really want to do it find the values of $x_1,x_2,x_3$ such that you get back $x_1,x_2,x_3$ when you change the basis. So $y_1=x_2$, $y_2=x_2$ and $y_3=x_3$ which you then put into your system of equations: $$ x_1=2x_1-x_2-x_3\\ x_2=-x_2 \\ x_3=2x_2+x_3 $$ So from that we can see $x_2=0$ so then we are just solving: $$ x_1=2x_1-x_3\\ x_3=x_3 $$ which simplifies to $x_1=x_3$ so we can represent the solution as any vector in the form $(t,0,t)$ for all $t\in \mathbb{R}$. Which you can quickly check by putting in your system of equations is invariant under the base change.

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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.

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