An algebra problem by Rocco Dalto

Algebra Level pending

Let \( f: \mathbb R^4 \rightarrow \mathbb P_{3} \) be linear transform defined by: \(\)

\[ f \left( \begin{array}{cccc} p_{1} \\ p_{2} \\ p_{3} \\ p_{4} \\ \end{array} \right) = p1 + p2 * x + p3 * x^2 + p4 * x^3 \]

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Let \[ A = \{ \left( \begin{array}{cccc} 1 \\ 2 \\ 3 \\ 4 \\ \end{array} \right), \left( \begin{array}{cccc} 1 \\ -1 \\ 2 \\ 4 \\ \end{array} \right), \left( \begin{array}{cccc} 1 \\ 1 \\ -1 \\ 2 \\ \end{array} \right), \left( \begin{array}{cccc} 1 \\ 4 \\ 2 \\ 3 \\ \end{array} \right) \} \] be a basis for \( \mathbb R^4 \).

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Let \[ B = \{ 1 + x - x^2 + x^3, 1 - x^2, x - 3 * x^2 - x^3, 1 + 2 * x - 3 * x^2 - 4 * x^3 \} \] be a basis for \( \mathbb P_{3} \).

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If \( M = [a_{ij}]_{4 \: x \: 4} \) represents the linear transform above find \( \displaystyle S = \sum_{i = 1}^{4} \sum_{j = 1}^{4} a_{i j}.\) \(\)

Express the result to four decimal places.

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If your interested you can write a program(in any language) for the General Case below and use it to find the above result.

General Case(for program)

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Let \( f: \mathbb R^n \rightarrow \mathbb P_{n - 1} \) be linear transform defined by: \(\)

\[ f \left( \begin{array}{cccccccccc} p_{1} \\ p_{2} \\ . \\ . \\ . \\ p_{j} \\ . \\ . \\ . \\ p_{n} \\ \end{array} \right) = \sum_{j = 1}^{n} p_{j} * x^{j - 1} \]

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Let \[ V_{q} = \left( \begin{array}{cccccccccc} p_{1q} \\ p_{2q} \\ . \\ . \\ . \\ p_{jq} \\ . \\ . \\ . \\ p_{nq} \\ \end{array} \right) \] \(\)

and \( A = \{V_{q}| (1 <= q <= n) \} \) be a basis for \( \mathbb R^n \)

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Let \[ W_{q} = \sum_{j = 1}^{n} p_{jq} * x^{j - 1} \] \(\)

and \(B = \{W_{q}| (1 <= q <= n) \} \) be a basis for \( \mathbb P_{n - 1} \)

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Write a program in any language to find the matrix \( M = [a_{ij}]_{n \: x \:n} \) representation of the general linear transform above and the sum \( \displaystyle S = \sum_{i = 1}^{n} \sum_{j = 1}^{n} a_{i j} \). \(\)

Make certain \( A \) and \( B \) are bases for \( \mathbb R^{n} \) and \( \mathbb P_{n - 1} \). \(\)

You can use the program written to find the matrix \( M = [a_{ij}]_{4 \: x \: 4} \) that represents the linear transform above and output \( \displaystyle S = \sum_{i = 1}^{4} \sum_{j = 1}^{4} a_{i j} \).

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Let \[X = \left( \begin{array}{cccc} 1 \\ 3\\ 7 \\ -5 \\ \end{array} \right) \]

To check the matrix \( M = [a_{ij}]_{4 \: x \: 4} \) found, first find \([f(X)]_B\) without using the matrix \( M = [a_{ij}]_{4 \: x \: 4} \), then find \([f(X)]_B\) using the matrix \( M = [a_{ij}]_{4 \: x \: 4} \).

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