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**arbolis** In a given basis $\displaystyle \{ \vec e _i \}$ of a vector space, a linear transformation and a given vector of this space are determinate by $\displaystyle \begin{bmatrix} 2 & 1 & 0 \\ 1&2&0 \\ 0 & 0 & 5 \end{bmatrix}$ and $\displaystyle \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$ (should be a column vector).

Find the matrix representation of the transformation and of the vector in a new basis such that the old one is represented by $\displaystyle \vec e _1 = \begin{bmatrix} 1&1&0 \end{bmatrix}$, $\displaystyle \vec e _2 = \begin{bmatrix} 1&-1&0 \end{bmatrix}$,$\displaystyle \vec e _3 = \begin{bmatrix} 0&0&1 \end{bmatrix}$ (they should be column vectors).

My attempt: I formed a matrix whose columns are the $\displaystyle \vec e_1$ ,$\displaystyle \vec e_2$ ,$\displaystyle \vec e_3$ and I found its inverse. Call them $\displaystyle S$ and $\displaystyle S^{-1}$.

Then I multiplied $\displaystyle S^{-1}AS$ where $\displaystyle A$ is the first matrix I wrote. As result, I obtained the following matrix $\displaystyle A'= \begin{bmatrix} 3&0&0 \\ 0&1&0 \\ 0&0&5 \end{bmatrix}$ which would be the matrix they asked for.

For the vector, I wasn't sure at all... I just multipled $\displaystyle A'$ by $\displaystyle \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$ and I obtained $\displaystyle \begin{bmatrix} 3 & 12 & 15 \end{bmatrix}$ which would be the vector they ask for.

Am I right?