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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