# 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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