# An algebra problem by Rocco Dalto

Algebra Level pending

Let $$\mathbb P_{2}$$ denote the set of all polynomials of degree two and $$\mathbb R^3$$ denote the set of all vectors in three space.



Let $$f: \mathbb P_{2} \rightarrow \mathbb R^3$$ be linear transform defined by

$f(p_{0} + p_{1} x + p_{2} x^2) = \left( \begin{array}{ccc} p_{0} + p_{1} \\ p_{1} + p_{2} \\ p_{0} + p_{2} \\ \end{array} \right)$

Let
$A = \{1 + x, x + x^2, x^2 \}$ be a basis for $$\mathbb P_{2}$$ 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 $$\mathbb R^3.$$.

Let the matrix $$M = [a_{ij}]_{3 \: x \: 3}$$ represent the linear transform above. 

If $$[p]_{A} = \left( \begin{array}{ccc} x_{1} \\ x_{2} \\ x_{3} \ \end{array} \right)$$, $$\: [f(p)]_{B} = \left( \begin{array}{ccc} 1 \\ 2 \\ 3 \ \end{array} \right)$$, $$\: S = \sum_{j = 1}^{3} x_{j}$$, $$\: p = b_{0} + b_{1} x + b_{2} x^2$$, and $$\: T = \sum_{j = 0}^{2} b_{j}$$, 

Find: $$S + T$$.



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