An algebra problem by Rocco Dalto

Algebra Level pending

Let \( f: \mathbb R^2 \rightarrow \mathbb R^3 \) be linear transform defined by: \(\)

\[ f \left( \begin{array}{ccc} x_{1} \\ x_{2} \\ \end{array} \right) = \left( \begin{array}{ccc} 3 x_{1} - x_{2}\\ x_{2} \\ x_{1} + x_{2} \end{array} \right)\]

and \[ A = \{ \left( \begin{array}{ccc} -1 \\ 1 \\ \end{array} \right), \left( \begin{array}{ccc} 2 \\ 1 \\ \end{array} \right) \} \] be basis for \( \mathbb R^2 \) and \[ B = \left \{ \left( \begin{array}{ccc} 1 \\ -1 \\ 1 \\ \end{array} \right),
\left( \begin{array}{ccc} -1 \\ 1 \\ 1 \ \end{array} \right), \left( \begin{array}{ccc} 1 \\ 0 \\ 1 \ \end{array} \right) \right \} \] be a basis for \( \mathbb R^3. \).

Find a matrix representation for the linear transform above.

If \( \: \textbf{X} = \left( \begin{array}{ccc} 1 \\ 2 \\ \end{array} \right) \in \mathbb R^2 \) and \( \textbf{[f(X)]}_{\textbf{B}} = \left( \begin{array}{ccc} \lambda_1 \\ \lambda_2 \\ \lambda_3 \ \end{array} \right) \), \(\)

Find: \( \lambda_1 + \lambda_2 + \lambda_3 \).

\(\)

Note: Of course you can find \( \textbf{[f(X)]}_{\textbf{B}} = \left( \begin{array}{ccc} \lambda_1 \\ \lambda_2 \\ \lambda_3 \ \end{array} \right) \) without finding the matrix, but my intention was to find the matrix.

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