# An algebra problem by Rocco Dalto

Algebra Level pending

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

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



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 $$\mathbb R^4$$.



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 $$\mathbb P_{3}$$.





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.



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)



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

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



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 $$\mathbb R^n$$



Let $W_{q} = \sum_{j = 1}^{n} p_{jq} * x^{j - 1}$ 

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



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 $$\displaystyle S = \sum_{i = 1}^{4} \sum_{j = 1}^{4} a_{i j}$$.



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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