Change of Bases Problem

Algebra Level pending

Consider the matrix A: \(\mathbb{R}^{3} \rightarrow \mathbb{R}^{2} \) Given by,

\[ \begin{bmatrix}1&3&5 \\ 4&1&-1 \end{bmatrix} \]

Let \( e_{1} \, e_{2} \, e_{3} \) denote the standard basis and consider a new basis \( \{v_{1} , v_{2} , v_{3} \} \) for the domain and \( \{w_{1} , w_{2} \} \) for the co-domain which are defined in terms of the standard basis as,

\[ \{ v_{1} , v_{2} , v_{3} \}= \{ e_{1}+e_{2}+e_{3},e_{2}+e_{3}, e_{3} \} \] \[ \{ w_{1} , w_{2} \}= \{ e_{1}-e_{2}, e_{2} \} \]

What is equivalent representation of L in these new bases?

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