# An algebra problem by Rocco Dalto

Algebra Level pending

Let $$f: \mathbb A_{2 X 2} \rightarrow \mathbb P_3$$ be linear transform defined by:

$f( \begin{vmatrix}{p_{1}} && {p_{2}} \\ {p_{3}} && {p_{4}}\end{vmatrix}) = p1 + p2 * x + p3 * x^2 + p4 * x^3$



Let $A = \{ \begin{vmatrix}{1} && {2} \\ {3} && {4}\end{vmatrix}, \begin{vmatrix}{-2} && {1} \\ {5} && {3}\end{vmatrix}, \begin{vmatrix}{4} && {-2} \\ {1} && {7}\end{vmatrix}, \begin{vmatrix}{7} && {6} \\ {4} && {3}\end{vmatrix} \}$ be a basis for $$\mathbb A_{2 X 2}$$



and,

$B = \{ 2 - x + 4 * x^2 + 3 * x^3, 5 + 2 * x + x^2 + 4 * x^3, 1 - x + x^2 + 2 * x^3, 2 + x^2 - x^3 \}$ 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)



For this case I wrote the matrix in an unconventional manner. 

Let $$f: \mathbb A_{n X n} \rightarrow \mathbb P_{n^2 - 1}$$ be linear transform defined by:

$f( \begin{vmatrix}{x_{1}} && {x_{2}} && {...} && {x_{n}} \\ {x_{n + 1}} && {x_{n + 2}} && {...} && {x_{2 * n}} \\ {...} && {...} && {...} \\ {x_{(j - 1) * n + 1}} && {x_{(j - 1) * n + 2}} && {...} && {x_{j * n}} \\ {...} && {...} && {...} \\ {x_{(n - 1) * n + 1}} && {x_{(n - 1) * n + 2}} && {...} && {x_{n^2}} \\ \end{vmatrix}) = \sum_{j = 1}^{n^2} p_{j} * x^{j - 1}$



Let $V_{q} = \begin{vmatrix}{x_{1q}} && {x_{2q}} && {...} && {x_{nq}} \\ {x_{(n + 1)q}} && {x_{(n + 2)q}} && {...} && {x_{(2 * n)q}} \\ {...} && {...} && {...} \\ {x_{((j - 1) * n + 1)q}} && {x_{((j - 1) * n + 2)q}} && {...} && {x_{(j * n)q}} \\ {...} && {...} && {...} \\ {x_{((n - 1) * n + 1)q}} && {x_{((n - 1) * n + 2)q}} && {...} && {x_{(n^2)q}} \\ \end{vmatrix}$



and $$A = \{V_{q}|(1 <= q <= n^2)\}$$ be a basis for $$\mathbb A_{n X n}$$.



Let $W_{q} = \sum_{j = 1}^{n^2} p_{jq} * x^{j - 1}$ 

and $$B = \{W_{q}| (1 <= q <= n^2) \}$$ be a basis for $$\mathbb P_{n^2 - 1}$$



Write a program in any language to find the matrix $$M = [a_{ij}]_{n^2 \: x \:n^2}$$ representation of the general linear transform above and the sum $$\displaystyle S = \sum_{i = 1}^{n^2} \sum_{j = 1}^{n^2} a_{i j}$$. 

Make certain $$A$$ and $$B$$ are bases for $$\mathbb A_{n X n}$$ and $$\mathbb P_{n^2 - 1}$$. 

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 $X = \begin{vmatrix}{3} && {-2} \\ {5} && {7}\end{vmatrix}$

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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