An algebra problem by Rocco Dalto

Algebra Level 4

Let f : A 2 X 2 → R 4 f: \mathbb A_{2 X 2} \rightarrow \mathbb R^4 be linear transform defined by:

f ( ∣ x 1 x 2 x 3 x 4 ∣ ) = ( x 1 − x 2 + 2 ∗ x 3 + 4 ∗ x 4 2 ∗ x 1 − x 2 + 4 ∗ x 3 + 3 ∗ x 4 4 ∗ x 1 − 3 ∗ x 2 + 2 ∗ x 3 + x 4 5 ∗ x 1 − x 2 + 2 ∗ x 3 + 2 ∗ x 4 ) f( \begin{vmatrix}{x_{1}} && {x_{2}} \\ {x_{3}} && {x_{4}}\end{vmatrix}) = \left( \begin{array}{cccc} x_{1} - x_{2} + 2 * x_{3} + 4 * x_{4} \\ 2 * x_{1} - x_{2} + 4 * x_{3} + 3 * x_{4} \\ 4 * x_{1} - 3 * x_{2} + 2 * x_{3} + x_{4} \\ 5 * x_{1} - x_{2} + 2 * x_{3} + 2 * x_{4} \\ \end{array} \right)

Let A = { ∣ 1 2 3 4 ∣ , ∣ − 2 1 5 3 ∣ , ∣ 4 − 2 1 7 ∣ , ∣ 7 6 4 3 ∣ } 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 A 2 X 2 \mathbb A_{2 X 2}

and,

B = { ( 2 − 1 4 3 ) , ( 5 2 1 4 ) , ( 1 − 1 1 2 ) , ( 2 0 1 − 1 ) } B = \{ \left( \begin{array}{cccc} 2 \\ -1 \\ 4 \\ 3 \\ \end{array} \right), \left( \begin{array}{cccc} 5 \\ 2 \\ 1 \\ 4 \\ \end{array} \right), \left( \begin{array}{cccc} 1 \\ -1 \\ 1 \\ 2 \\ \end{array} \right), \left( \begin{array}{cccc} 2 \\ 0 \\ 1 \\ -1 \\ \end{array} \right) \} be a basis for R 4 \mathbb R^4

If M = [ a i j ] 4 x 4 M = [a_{ij}]_{4 \: x \: 4} represents the linear transform above find S = ∑ i = 1 4 ∑ j = 1 4 a i j . \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 : A n X n → R n 2 f: \mathbb A_{n X n} \rightarrow \mathbb R^{n^2} be linear transform defined by:

f ( ∣ 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 ∣ ) = ( C 11 ∗ x 1 + C 12 ∗ x 2 + . . . + C 1 ∗ n 2 ∗ x n 2 C 21 ∗ x 1 + C 22 ∗ x 2 + . . . + C 2 ∗ n 2 ∗ x n 2 . . . C j 1 ∗ x 1 + C j 2 ∗ x 2 + . . . + C j ∗ n 2 ∗ x n 2 . . . C n 2 1 ∗ x 1 + C n 2 2 ∗ x 2 + . . . + C n 2 n 2 ∗ x n 2 ) 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}) = \left( \begin{array}{cccc} C_{11} * x_{1} + C_{12} * x_{2} + ... + C_{1 * n^2} * x_{n^2} \\ C_{21} * x_{1} + C_{22} * x_{2} + ... + C_{2 * n^2} * x_{n^2} \\ ... \\ C_{j1} * x_{1} + C_{j2} * x_{2} + ... + C_{j * n^2} * x_{n^2} \\ ... \\ C_{n^21} * x_{1} + C_{n^22} * x_{2} + ... + C_{n^2n^2} * x_{n^2} \\ \end{array} \right)

Let V q = ∣ x 1 q x 2 q . . . x n q 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 ∣ 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 ) } A = \{V_{q}|(1 <= q <= n^2)\} be a basis for A n X n \mathbb A_{n X n} .

Let W q = ( a 1 q a 2 q . . . a j q . . . a ( n 2 ) q ) W_{q} = \left( \begin{array}{cccccc} a_{1q} \\ a_{2q} \\ ... \\ a_{jq} \\ ... \\ a_{(n^2)q} \\ \end{array} \right)

and B = { W q ∣ ( 1 < = q < = n 2 ) } B = \{W_{q}|(1 <= q <= n^2)\} be a basis for R n 2 \mathbb R^{n^2} .

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

Make certain A A and B B are bases for A n X n \mathbb A_{n X n} and R n 2 \mathbb R^{n^2} .

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

Let X = ∣ 3 − 2 5 7 ∣ X = \begin{vmatrix}{3} && {-2} \\ {5} && {7}\end{vmatrix}

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


The answer is -44.0678.

Rocco Dalto by Rocco Dalto , 67, USA

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1 solution

Rocco Dalto
Apr 25, 2017

After doing the problem I supply the program I wrote for the general case which generated the output for this specific problem.

Since f : A 2 X 2 → R 4 f: \mathbb A_{2 X 2} \rightarrow \mathbb R^{4} is a linear transform, for each integer j ∋ ( 1 < = j < = 4 ) j \ni (1 <= j <= 4)

f ( ∣ x 1 j x 2 j x 3 j x 4 j ∣ ) = α 1 j ∗ ( 2 − 1 4 3 ) + α 2 j ∗ ( 5 2 1 4 ) + α 3 j ∗ ( 1 − 1 1 2 ) + α 4 j ∗ ( 2 0 1 − 1 ) = f( \begin{vmatrix}{x_{1j}} && {x_{2j}} \\ {x_{3j}} && {x_{4j}}\end{vmatrix}) = \alpha_{1j} * \left( \begin{array}{cccc} 2 \\ -1 \\ 4 \\ 3 \\ \end{array} \right) + \alpha_{2j} * \left( \begin{array}{cccc} 5 \\ 2 \\ 1 \\ 4 \\ \end{array} \right) + \alpha_{3j} * \left( \begin{array}{cccc} 1 \\ -1 \\ 1 \\ 2 \\ \end{array} \right) + \alpha_{4j} * \left( \begin{array}{cccc} 2 \\ 0 \\ 1 \\ -1 \\ \end{array} \right) = [ 2 5 1 2 − 1 2 − 1 0 4 1 1 1 3 4 2 − 1 ] ∗ ∣ α 1 j α 2 j α 3 j α 4 j ∣ \begin{bmatrix}{2} && {5} && {1} && {2} \\ {-1} && {2} && {-1} && {0} \\ {4} && {1} && {1} && {1} \\ {3} && {4} && {2} && {-1}\end{bmatrix} * \begin{vmatrix}{\alpha_{1j}} \\{\alpha_{2j}} \\ {\alpha_{3j}} \\ {\alpha_{4j}}\end{vmatrix}

where,

f ( ∣ 1 2 3 4 ∣ ) = ( 21 24 8 17 ) f( \begin{vmatrix}{1} && {2} \\ {3} && {4}\end{vmatrix}) = \left( \begin{array}{cccc} 21 \\ 24 \\ 8 \\ 17 \\ \end{array} \right)

f ( ∣ − 2 1 5 3 ∣ ) = ( 19 24 2 5 ) f( \begin{vmatrix}{-2} && {1} \\ {5} && {3}\end{vmatrix}) = \left( \begin{array}{cccc} 19 \\ 24 \\ 2 \\ 5 \\ \end{array} \right)

f ( ∣ 4 − 2 1 7 ∣ ) = ( 36 35 31 38 ) f( \begin{vmatrix}{4} && {-2} \\ {1} && {7}\end{vmatrix}) = \left( \begin{array}{cccc} 36 \\ 35 \\ 31 \\ 38 \\ \end{array} \right)

f ( ∣ 7 6 4 3 ∣ ) = ( 21 33 21 43 ) f( \begin{vmatrix}{7} && {6} \\ {4} && {3}\end{vmatrix}) = \left( \begin{array}{cccc} 21 \\ 33 \\ 21 \\ 43 \\ \end{array} \right)

Using the above we can set up the augmented matrix below to solve for the

four 4 X 4 systems of equations.

[ 2.0000 5.0000 1.0000 2.0000 21.0000 19.0000 36.0000 21.0000 − 1.0000 2.0000 − 1.0000 0.0000 24.0000 24.0000 35.0000 33.0000 4.0000 1.0000 1.0000 1.0000 8.0000 2.0000 31.0000 21.0000 3.0000 4.0000 2.0000 − 1.0000 17.0000 5.0000 38.0000 43.0000 ] \left[ \begin{array}{cccc|cccc} 2.0000 & 5.0000 & 1.0000 & 2.0000 & 21.0000 & 19.0000 & 36.0000 & 21.0000\\ -1.0000 & 2.0000 & -1.0000 & 0.0000 & 24.0000 & 24.0000 & 35.0000 & 33.0000\\ 4.0000 & 1.0000 & 1.0000 & 1.0000 & 8.0000 & 2.0000 & 31.0000 & 21.0000\\ 3.0000 & 4.0000 & 2.0000 & -1.0000 & 17.0000 & 5.0000 & 38.0000 & 43.0000\\ \ \end{array} \right]

0.5000 ∗ R o w 1 → 0.5000 * Row_{1} \rightarrow

[ 1.0000 2.5000 0.5000 1.0000 10.5000 9.5000 18.0000 10.5000 − 1.0000 2.0000 − 1.0000 0.0000 24.0000 24.0000 35.0000 33.0000 4.0000 1.0000 1.0000 1.0000 8.0000 2.0000 31.0000 21.0000 3.0000 4.0000 2.0000 − 1.0000 17.0000 5.0000 38.0000 43.0000 ] \left[ \begin{array}{cccc|cccc} 1.0000 & 2.5000 & 0.5000 & 1.0000 & 10.5000 & 9.5000 & 18.0000 & 10.5000 \\ -1.0000 & 2.0000 & -1.0000 & 0.0000 & 24.0000 & 24.0000 & 35.0000 & 33.0000\\ 4.0000 & 1.0000 & 1.0000 & 1.0000 & 8.0000 & 2.0000 & 31.0000 & 21.0000\\ 3.0000 & 4.0000 & 2.0000 & -1.0000 & 17.0000 & 5.0000 & 38.0000 & 43.0000\\ \ \end{array} \right]

1.0000 ∗ R O W 1 + R O W 2 1.0000 * ROW_{1} + ROW_{2}

− 4.0000 ∗ R O W 1 + R O W 3 -4.0000 * ROW_{1} + ROW_{3}

− 3.0000 ∗ R O W 1 + R O W 4 → -3.0000 * ROW_{1} + ROW_{4} \rightarrow

[ 1.0000 2.5000 0.5000 1.0000 10.5000 9.5000 18.0000 10.5000 0.0000 4.5000 − 0.5000 1.0000 34.5000 33.5000 53.0000 43.5000 0.0000 − 9.0000 − 1.0000 − 3.0000 − 34.0000 − 36.0000 − 41.0000 − 21.0000 0.0000 − 3.5000 0.5000 − 4.0000 − 14.5000 − 23.5000 − 16.0000 11.5000 ] \left[ \begin{array}{cccc|cccc} 1.0000 & 2.5000 & 0.5000 & 1.0000 & 10.5000 & 9.5000 & 18.0000 & 10.5000 \\ 0.0000 & 4.5000 & -0.5000 & 1.0000 & 34.5000 & 33.5000 & 53.0000 & 43.5000 \\ 0.0000 & -9.0000 & -1.0000 & -3.0000 & -34.0000 & -36.0000 & -41.0000 & -21.0000 \\ 0.0000 & -3.5000 & 0.5000 & -4.0000 & -14.5000 & -23.5000 & -16.0000 & 11.5000 \\ \ \end{array} \right]

0.2222 ∗ R o w 2 → 0.2222 * Row_{2} \rightarrow

[ 1.0000 2.5000 0.5000 1.0000 10.5000 9.5000 18.0000 10.5000 0.0000 1.0000 − 0.1111 0.2222 7.6667 7.4444 11.7778 9.6667 0.0000 − 9.0000 − 1.0000 − 3.0000 − 34.0000 − 36.0000 − 41.0000 − 21.0000 0.0000 − 3.5000 0.5000 − 4.0000 − 14.5000 − 23.5000 − 16.0000 11.5000 ] \left[ \begin{array}{cccc|cccc} 1.0000 & 2.5000 & 0.5000 & 1.0000 & 10.5000 & 9.5000 & 18.0000 & 10.5000\\ 0.0000 & 1.0000 & -0.1111 & 0.2222 & 7.6667 & 7.4444 & 11.7778 & 9.6667\\ 0.0000 & -9.0000 & -1.0000 & -3.0000 & -34.0000 & -36.0000 & -41.0000 & -21.0000\\ 0.0000 & -3.5000 & 0.5000 & -4.0000 & -14.5000 & -23.5000 & -16.0000 & 11.5000 \\ \ \end{array} \right]

− 2.5000 ∗ R O W 2 + R O W 1 -2.5000 * ROW_{2} + ROW_{1}

9.0000 ∗ R O W 2 + R O W 3 9.0000 * ROW_{2} + ROW_{3}

3.5000 ∗ R O W 2 + R O W 4 → 3.5000 * ROW_{2} + ROW_{4} \rightarrow

[ 1.0000 0.0000 0.7778 0.4444 − 8.6667 − 9.1111 − 11.4444 − 13.6667 0.0000 1.0000 − 0.1111 0.2222 7.6667 7.4444 11.7778 9.6667 0.0000 0.0000 − 2.0000 − 1.0000 35.0000 31.0000 65.0000 66.0000 0.0000 0.0000 0.1111 − 3.2222 12.3333 2.5556 25.2222 45.3333 ] \left[ \begin{array}{cccc|cccc} 1.0000 & 0.0000 & 0.7778 & 0.4444 & -8.6667 & -9.1111 & -11.4444 & -13.6667\\ 0.0000 & 1.0000 & -0.1111 & 0.2222 & 7.6667 & 7.4444 & 11.7778 & 9.6667\\ 0.0000 & 0.0000 & -2.0000 & -1.0000 & 35.0000 & 31.0000 & 65.0000 & 66.0000\\ 0.0000 & 0.0000 & 0.1111 & -3.2222 & 12.3333 & 2.5556 & 25.2222 & 45.3333\\ \ \end{array} \right]

− 0.5000 ∗ R o w 3 → -0.5000 * Row3 \rightarrow

[ 1.0000 0.0000 0.7778 0.4444 − 8.6667 − 9.1111 − 11.4444 − 13.6667 0.0000 1.0000 − 0.1111 0.2222 7.6667 7.4444 11.7778 9.6667 0.0000 0.0000 1.0000 0.5000 − 17.5000 − 15.5000 − 32.5000 − 33.0000 0.0000 0.0000 0.1111 − 3.2222 12.3333 2.5556 25.2222 45.3333 ] \left[ \begin{array}{cccc|cccc} 1.0000 & 0.0000 & 0.7778 & 0.4444 & -8.6667 & -9.1111 & -11.4444 & -13.6667\\ 0.0000 & 1.0000 & -0.1111 & 0.2222 & 7.6667 & 7.4444 & 11.7778 & 9.6667\\ 0.0000 & 0.0000 & 1.0000 & 0.5000 & -17.5000 & -15.5000 & -32.5000 & -33.0000\\ 0.0000 & 0.0000 & 0.1111 & -3.2222 & 12.3333 & 2.5556 & 25.2222 & 45.3333\\ \ \end{array} \right]

− 0.7778 ∗ R O W 3 + R O W 1 -0.7778 * ROW_{3} + ROW_{1}

0.1111 ∗ R O W 3 + R O W 2 0.1111 * ROW_{3} + ROW_{2}

− 0.1111 ∗ R O W 3 + R O W 4 → -0.1111 * ROW_{3} + ROW_{4} \rightarrow

[ 1.0000 0.0000 0.0000 0.0556 4.9444 2.9444 13.8333 12.0000 0.0000 1.0000 0.0000 0.2778 75.7222 5.7222 8.1667 6.0000 0.0000 0.0000 1.0000 0.5000 − 17.5000 − 15.5000 − 32.5000 − 33.0000 0.0000 0.0000 0.0000 − 3.2778 14.2778 4.2778 28.8333 49.0000 ] \left[ \begin{array}{cccc|cccc} 1.0000 & 0.0000 & 0.0000 & 0.0556 & 4.9444 & 2.9444 & 13.8333 & 12.0000\\ 0.0000 & 1.0000 & 0.0000 & 0.2778 & 7 5.7222 & 5.7222 & 8.1667 & 6.0000\\ 0.0000 & 0.0000 & 1.0000 & 0.5000 & -17.5000 & -15.5000 & -32.5000 & -33.0000\\ 0.0000 & 0.0000 & 0.0000 & -3.2778 & 14.2778 & 4.2778 & 28.8333 & 49.0000\\ \ \end{array} \right]

− 0.3051 ∗ R o w 4 → -0.3051 * Row_{4} \rightarrow

[ 1.0000 0.0000 0.0000 0.0556 4.9444 2.9444 13.8333 12.0000 0.0000 1.0000 0.0000 0.2778 5.7222 5.7222 8.1667 6.0000 0.0000 0.0000 1.0000 0.5000 − 17.5000 − 15.5000 − 32.5000 − 33.0000 0.0000 0.0000 0.0000 1.0000 − 4.3559 − 1.3051 − 8.7966 − 14.9492 ] \left[ \begin{array}{cccc|cccc} 1.0000 & 0.0000 & 0.0000 & 0.0556 & 4.9444 & 2.9444 & 13.8333 & 12.0000\\ 0.0000 & 1.0000 & 0.0000 & 0.2778 & 5.7222 & 5.7222 & 8.1667 & 6.0000\\ 0.0000 & 0.0000 & 1.0000 & 0.5000 & -17.5000 & -15.5000 & -32.5000 & -33.0000\\ 0.0000 & 0.0000 & 0.0000 & 1.0000 & -4.3559 & -1.3051 & -8.7966 & -14.9492\\ \ \end{array} \right]

− 0.0556 ∗ R O W 4 + R O W 1 -0.0556 * ROW_{4} + ROW_{1}

− 0.2778 ∗ R O W 4 + R O W 2 -0.2778 * ROW_{4} + ROW_{2}

− 0.5000 ∗ R O W 4 + R O W 3 → -0.5000 * ROW_{4} + ROW_{3} \rightarrow

[ 1.0000 0.0000 0.0000 0.0000 5.1864 3.0169 14.3220 12.8305 0.0000 1.0000 0.0000 0.0000 6.9322 6.0847 10.6102 10.1525 0.0000 0.0000 1.0000 0.0000 − 15.3220 − 14.8475 − 28.1017 − 25.5254 0.0000 0.0000 0.0000 1.0000 − 4.3559 − 1.3051 − 8.7966 − 14.9492 ] \left[ \begin{array}{cccc|cccc} 1.0000 & 0.0000 & 0.0000 & 0.0000 & 5.1864 & 3.0169 & 14.3220 & 12.8305\\ 0.0000 & 1.0000 & 0.0000 & 0.0000 & 6.9322 & 6.0847 & 10.6102 & 10.1525\\ 0.0000 & 0.0000 & 1.0000 & 0.0000 & -15.3220 & -14.8475 & -28.1017 & -25.5254\\ 0.0000 & 0.0000 & 0.0000 & 1.0000 & -4.3559 & -1.3051 & -8.7966 & -14.9492\\ \ \end{array} \right]

The desired matrix is:

M = [ 5.1864 3.0169 14.3220 12.8305 6.9322 6.0847 10.6102 10.1525 − 15.3220 − 14.8475 − 28.1017 − 25.5254 − 4.3559 − 1.3051 − 8.7966 − 14.9492 ] M = \begin{bmatrix}{5.1864} && {3.0169} && {14.3220} && {12.8305} \\ {6.9322} && {6.0847} && {10.6102} && {10.1525} \\ {-15.3220} && {-14.8475} && {-28.1017} && {-25.5254} \\ {-4.3559} && {-1.3051} && {-8.7966} && {-14.9492} \end{bmatrix}

and S = ∑ i = 1 4 ∑ j = 1 4 a i j = − 44.0678 S = \sum_{i = 1}^{4} \sum_{j = 1}^{4} a_{i j} = \boxed{-44.0678} .

Check [ f ( X ) ] B : [f(X)]_{B}:

Without using matrix M : M:

Let X = ∣ 3 − 2 5 7 ∣ X = \begin{vmatrix}{3} && {-2} \\ {5} && {7}\end{vmatrix}

⟹ \implies

f ( ∣ 3 − 2 5 7 ∣ ) = ( 43 49 35 41 ) = α 1 ∗ ( 2 − 1 4 3 ) + α 2 ∗ ( 5 2 1 4 ) + α 3 ∗ ( 1 − 1 1 2 ) + α 4 ∗ ( 2 0 1 − 1 ) = f( \begin{vmatrix}{3} && {-2} \\ {5} && {7}\end{vmatrix}) = \left( \begin{array}{cccc} 43 \\ 49 \\ 35 \\ 41 \\ \end{array} \right) = \alpha_{1} * \left( \begin{array}{cccc} 2 \\ -1 \\ 4 \\ 3 \\ \end{array} \right) + \alpha_{2} * \left( \begin{array}{cccc} 5 \\ 2 \\ 1 \\ 4 \\ \end{array} \right) + \alpha_{3} * \left( \begin{array}{cccc} 1 \\ -1 \\ 1 \\ 2 \\ \end{array} \right) + \alpha_{4} * \left( \begin{array}{cccc} 2 \\ 0 \\ 1 \\ -1 \\ \end{array} \right) = [ 2 5 1 2 − 1 2 − 1 0 4 1 1 1 3 4 2 − 1 ] \begin{bmatrix}{2} && {5} && {1} && {2} \\ {-1} && {2} && {-1} && {0} \\ {4} && {1} && {1} && {1} \\ {3} && {4} && {2} && {-1}\end{bmatrix} * ∣ α 1 α 2 α 3 α 4 ∣ \begin{vmatrix}{\alpha_{1}} \\{\alpha_{2}} \\ {\alpha_{3}} \\ {\alpha_{4}}\end{vmatrix}

Using the above we can set up the augmented matrix:

[ 2.0000 5.0000 1.0000 2.0000 43.0000 − 1.0000 2.0000 − 1.0000 0.0000 49.0000 4.0000 1.0000 1.0000 1.0000 35.0000 3.0000 4.0000 2.0000 − 1.0000 41.0000 ] \left[ \begin{array}{cccc|c} 2.0000 & 5.0000 & 1.0000 & 2.0000 & 43.0000\\ -1.0000 & 2.0000 & -1.0000 & 0.0000 & 49.0000\\ 4.0000 & 1.0000 & 1.0000 & 1.0000 & 35.0000\\ 3.0000 & 4.0000 & 2.0000 & -1.0000 & 41.0000\\ \ \end{array} \right]

Using row operations on the above augmented matrix we obtain:

[ 1.0000 0.0000 0.0000 0.0000 17.9492 0.0000 1.0000 0.0000 0.0000 13.7458 0.0000 0.0000 1.0000 0.0000 − 39.4576 0.0000 0.0000 0.0000 1.0000 − 11.0847 ] \left[ \begin{array}{cccc|c} 1.0000 & 0.0000 & 0.0000 & 0.0000 & 17.9492\\ 0.0000 & 1.0000 & 0.0000 & 0.0000 & 13.7458\\ 0.0000 & 0.0000 & 1.0000 & 0.0000 & -39.4576\\ 0.0000 & 0.0000 & 0.0000 & 1.0000 & -11.0847\\ \ \end{array} \right]

⟹ \implies

[ f ( X ) ] B = ( 17.9492 13.7458 − 39.4576 − 11.0847 ) [f(X)]_{B} = \left( \begin{array}{ccccc} 17.9492\\ 13.7458\\ -39.4576\\ -11.0847\\ \end{array} \right)

Using matrix M M :

Using the set of matrices A A and X = ∣ 3 − 2 5 7 ∣ X = \begin{vmatrix}{3} && {-2} \\ {5} && {7}\end{vmatrix} we set up the system:

[ 1.0000 − 2.0000 4.0000 7.0000 3.0000 2.0000 1.0000 − 2.0000 6.0000 − 2.0000 3.0000 5.0000 1.0000 4.0000 5.0000 4.0000 3.0000 7.0000 3.0000 7.0000 ] \left[ \begin{array}{cccc|c} 1.0000 & -2.0000 & 4.0000 & 7.0000 & 3.0000\\ 2.0000 & 1.0000 & -2.0000 & 6.0000 & -2.0000\\ 3.0000 & 5.0000 & 1.0000 & 4.0000 & 5.0000\\ 4.0000 & 3.0000 & 7.0000 & 3.0000 & 7.0000\\ \end{array} \right]

Solving the system we obtain:

[ X ] A = [ − 1.8274 1.5112 1.2130 0.4283 ] [X]_A = \left[ \begin{array}{ccccc} -1.8274\\ 1.5112\\ 1.2130\\ 0.4283\\ \end{array} \right]

⟹ \implies

[ f ( p ) ] B = M ∗ [ X ] A = [ 5.1864 3.0169 14.3220 12.8305 6.9322 6.0847 10.6102 10.1525 − 15.3220 − 14.8475 − 28.1017 − 25.5254 − 4.3559 − 1.3051 − 8.7966 − 14.9492 ] ∗ [ − 1.8274 1.5112 1.2130 0.4283 ] = ( 17.9492 13.7458 − 39.4576 − 11.0847 ) [f(p)]_B = M * [X]_A = \begin{bmatrix}{5.1864} && {3.0169} && {14.3220} && {12.8305} \\ {6.9322} && {6.0847} && {10.6102} && {10.1525} \\ {-15.3220} && {-14.8475} && {-28.1017} && {-25.5254} \\ {-4.3559} && {-1.3051} && {-8.7966} && {-14.9492} \end{bmatrix} * \left[ \begin{array}{ccccc} -1.8274\\ 1.5112\\ 1.2130\\ 0.4283\\ \end{array} \right] = \left( \begin{array}{ccccc} 17.9492\\ 13.7458\\ -39.4576\\ -11.0847\\ \end{array} \right)

I wrote the program in Free Pascal.

program matrixrepresentaionoflineartransform_4;

{$R-}

{restricted to f:Rm --> Anxn}

uses crt; 

                                {self contained for brillant}

const maxnum = 100;

type matrix = array[1 .. maxnum,1 .. maxnum] of real;

var 

coeff,a,b,c,d,sol,prod,e:matrix;

n,l,p,v:integer;

mywork3:text;

deta,detb,product:real;

f:matrix;{for me only-check}

function intpower(base,exponent:integer):longint;

var product,k:longint;

begin{intpower}

product:= 1; 

 for k:= 1 to exponent do

      product:= product * base;



 INTpower:= product;

end;{intpower}

procedure Getsize;

begin

writeln('Enter size of matrix');

readln(n);

end;

procedure getcoefficients;

var j,k,p:integer;

begin

for j:= 1 to intpower(n,2) do

begin

write('Enter coefficients of row',j,' :');

for k:= 1 to intpower(n,2) do

begin

read(coeff[j,k]);

end;

end;

end;

procedure Getmatrices;

var j,k,q,p:integer; 

sum:real;

myfile:text;

begin

assign(myfile,'holdmatrix3.txt');

rewrite(myfile);

writeln('To Enter ', intpower(n,2), ' basis vectors for A',n,'X',n,' : ');

for q:= 1 to intpower(n,2) do

begin

write('Enter matrix ', q, ' on one line: ');

for k:= 1 to intpower(n,2) do

begin

read(b[k,q]);

end;

end;

for q:= 1 to intpower(n,2) do

begin

for j:= 1 to intpower(n,2) do

begin

sum:= 0;

for k:= 1 to intpower(n,2) do

begin

sum:= sum + coeff[j,k] * b[k,q];

end;

write(myfile,sum);

end;

end;

writeln('Enter basis vectors in R',intpower(n,2),':');

for j:= 1 to intpower(n,2) do

begin

write('Enter enter vector',j,': ');

for k:= 1 to intpower(n,2) do

begin

read(a[k,j]);

end;

writeln;

end;

reset(myfile);

for j:= 1 to intpower(n,2) do

begin

for k:= 1 to intpower(n,2) do

begin

read(myfile,a[k,intpower(n,2) + j]);

end;

end;

writeln;

writeln(' Solve the ',intpower(n,2),' X ',intpower(n,2),' system using the augmented matrix below:');

writeln;

for j:= 1 to intpower(n,2) do

begin

for k:= 1 to 2 * intpower(n,2) do

begin

write(a[j,k]:0:4,' ');

end;

writeln;

end;

close(myfile);

for j:= 1 to intpower(n,2) do

begin

for k:= 1 to 2 * intpower(n,2) do

begin

write(mywork3,a[j,k]:0:4,' ');

end;

writeln(mywork3);

end;

end;

{For check bases} {begin}

procedure hold(var c:matrix; e:matrix; m:integer);

var k,j:integer;

begin

for k:= 1 to m do

begin

for j:= 1 to m do

begin

c[k,j]:= e[k,j];

end;

end;

end;

{Determinants-To check Bases} procedure switch2(var c:matrix; l,p,n:integer);

var q:integer; 

z:real;

begin{switch}

for q:= 1 to n do

begin{m} 

   z:= c[l,q];

   c[l,q]:= c[p,q];

   c[p,q]:= z;

end;{m}

end;{switch}

procedure safeguard2(var c:matrix; l:integer; var p:integer; n:integer; var v:integer; var product:real);

var q:integer;

begin{safeguard}

v:=0;

if (c[l,l] = 0) then

begin{then}

v:= 1;

q:= l;

while ((q + 1) <= n) and (v = 1) do

begin{loop}

if c[q + 1,l] <> 0 then

begin{then} 

        p:= q + 1;

        switch2(c,l,p,n);

        v:= 0;

        PRODUCT:= -1 * PRODUCT;

end;{then}

q:= q + 1;

end;{loop}

end;{then}

end;{safeguard}

procedure operation11(var a:matrix; l,n:integer; var product:real);

var k,q:integer; 

x:real;

s:matrix;

begin{operation1}

for q:= 1 to n do

begin{q}

s[l,q]:= a[l,q]/a[l,l];

end;{q}

PRODUCT:= PRODUCT * (1/A[L,L]);

for q:= 1 to n do

begin{q}

a[l,q]:= s[l,q];

end;{q}

    end;{operation1}

procedure operation21(var a:matrix; l,n:integer);

var k,r,q:integer; 

x:real;

s:matrix;

begin{op2}

for q:= 1 to n do

begin{q}

if q <> l then

begin{then}

x:= -1 * a[q,l];

for r:= 1 to n do

begin{r}

s[q,r]:= x * a[l,r] + a[q,r];

end;{r}

for r:= 1 to n do

begin{r}

a[q,r]:= s[q,r];

end;{r}

end;{then}

end;{q}

end;{op2}

FUNCTION DETERMINANT(a:matrix; n:integer; PRODUCT:real):real;

var j:integer; 

DET,DIAGONAL:real;

begin{DETERMINANT}

DIAGONAL:= 1;

FOR j:= 1 TO N DO

BEGIN{LOOP}

DIAGONAL:= DIAGONAL * (a[j,j]);

END;{LOOP}

IF DIAGONAL = 1 THEN

DET:= 1/PRODUCT

ELSE

DET:= 0;

DETERMINANT:= DET;

END;{DETERMINANT}

{end-check bases}

procedure switch(var c:matrix; l,p:integer);

var q:integer; 

z:real;

begin{switch}

for q:= 1 to 2 * intpower(n,2) do

begin{m} 

   z:= c[l,q];

   c[l,q]:= c[p,q];

   c[p,q]:= z;

end;{m}

end;{switch}

procedure safeguard(var c:matrix; l:integer; var p:integer; var v:integer);

var q:integer;

begin{safeguard}

v:=0;

if (c[l,l] = 0) then

begin{then}

v:= 1;

q:= l;

while ((q + 1) <= 2 * intpower(n,2)) and (v = 1) do

begin{loop}

if c[q + 1,l] <> 0 then

begin{then} 

        p:= q + 1;

        switch(c,l,p);

        v:= 0;

end;{then}

q:= q + 1;

end;{loop}

end;{then}

end;{safeguard}

procedure operation1(var a:matrix; l:integer);

var q,k:integer; 

s:matrix;

x:real;

                   {s is each transformed matrix}

begin{op1}

for q:= 1 to 2 * intpower(n,2) do

begin{m}

s[l,q]:= a[l,q]/a[l,l];

end;{m}

x:= 1/(a[l,l]);

for q:= 1 to 2 * intpower(n,2) do

begin{m}

a[l,q]:= s[l,q];

end;{m}

writeln(mywork3, x:0:4,' * Row',l);

writeln(mywork3);

for q:= 1 to intpower(n,2) do

begin

for k:= 1 to 2 * intpower(n,2) do

begin

write(mywork3, a[q,k]:0:4,' ');

end;

writeln(mywork3);

end;

writeln(mywork3);

end;{op1}

procedure operation2(var a:matrix; l:integer);

var q,r,k:integer; 

x:real;

s:matrix;

begin{op2}

for q:= 1 to intpower(n,2) do

begin{q}

if q <> l then

begin{then}

x:= a[q,l];

for r:= 1 to 2 * intpower(n,2) do

begin{m}

s[q,r]:= -1 * x * a[l,r] + a[q,r];

end;{m}

for r:= 1 to 2 * intpower(n,2) do

begin{m}

a[q,r]:= s[q,r];

end;{m}

writeln(mywork3, -1 * x:0:4,' * ROW ', l,' + ROW',q);

end;{then}

end;{q}

writeln(mywork3);

for q:= 1 to intpower(n,2) do

begin

for k:= 1 to 2 * intpower(n,2) do

begin

write(mywork3, a[q,k]:0:4,' ');

end;

writeln(mywork3);

end;

writeln(mywork3);

end;{op2}

PROCEDURE SHOWDATA;

VAR J,k:INTEGER; 

sum:extended;

myfile2:text;

BEGIN

assign(myfile2,'holdsolution3.txt');

rewrite(myfile2);

writeln;

writeln('Matrix Representation of linear transformation f: A',n,' X ',n,' ---> R',intpower(n,2),' : ');

writeln('------------------------------------------------------------------------------------------------------------');

writeln;

for j:= 1 to intpower(n,2) do

begin

for k:= intpower(n,2) + 1 to 2 * intpower(n,2) do

begin

write(myfile2,a[j,k],' ');

end;

writeln(myfile2);

end;

reset(myfile2);

for j:= 1 to intpower(n,2) do

begin

for k:= 1 to intpower(n,2) do

begin

read(myfile2,sol[j,k]);

end;

end;

close(myfile2);

for j:= 1 to intpower(n,2) do

begin

for k:= 1 to intpower(n,2) do

begin

write(sol[j,k]:0:4,' ');

end;

writeln;

end;

{For brillant only}

sum:= 0;

for j:= 1 to intpower(n,2) do

begin

for k:= 1 to intpower(n,2) do

begin

sum:= sum + sol[j,k];

end;

end;

writeln;

writeln('Sum of elements of matrix = ',sum:0:4);

END;

{Last check [f(x)]B } {just for me-begin}

procedure switch3(var c:matrix; l,p,n:integer);

var r:integer;

z:real;

begin{switch3}

for r:= 1 to n + 1 do

begin{m} 

   z:= c[l,r];

   c[l,r]:= c[p,r];

   c[p,r]:= z;

end;{m}

end;{switch3}

procedure safeguard3(var c:matrix; l:integer; var p:integer; n:integer; var v:integer);

var r:integer;

begin{safeguard3}

v:=0;

if (c[l,l] = 0) then

begin{then}

v:= 1;

r:= l;

while ((r + 1) <= n) and (v = 1) do

begin{loop}

if c[r + 1,l] <> 0 then

begin{then} 

        p:= r + 1;

        switch3(c,l,p,n);

        v:= 0;

end;{then}

r:= r + 1;

end;{loop}

end;{then}

end;{safeguard3}

procedure operation13(var a:matrix; l,n:integer);

var r,k:integer; 

s:matrix;

x:real;

                  {s is each transformed matrix}

begin{op13}

for r:= 1 to n + 1 do

begin{r}

s[l,r]:= a[l,r]/a[l,l];

end;{r}

x:= 1/(a[l,l]);

for r:= 1 to n + 1 do

begin{m}

a[l,r]:= s[l,r];

end;{r}

end;{op13}

procedure operation23(var a:matrix; l,n:integer);

var q,r:integer; 

x:real;

s:matrix;

begin{op23}

for q:= 1 to n do

begin{q}

if q <> l then

begin{then}

x:= a[q,l];

for r:= 1 to n + 1 do

begin{r}

s[q,r]:= -1 * x * a[l,r] + a[q,r];

end;{r}

for r:= 1 to n + 1 do

begin{r}

a[q,r]:= s[q,r];

end;{r}

end;{then}

end;{q}

end;{op23}

procedure solution(y:matrix; n:integer);

var q:integer;

begin{solution} 

for q:= 1 to n do

begin{q}

writeln(mywork3,y[q,n + 1]:0:4,'   ');

end;{q}

writeln(mywork3);

end;{solution}

procedure matrixmult(a,b:matrix; var summat:matrix);

var j,k,s:integer; 

   sum:real;

begin{matrixmult}

for j:= 1 to intpower(n,2) do

begin{j}

for k:= 1 to 1 do

begin{k} 

     sum:= 0;

for s:= 1 to intpower(n,2) do

begin{s} 

     sum:= sum + a[j,s] * b[s,k];

end;{s} 

      summat[j,k]:= sum;

              end;{k}

writeln(mywork3,sum:0:4);

end;{j}

writeln(mywork3);

end;{matrixmult}

\()

procedure check; t ype arraytype = array[1 .. maxnum] of real;

var p,j,k:integer; 

x:arraytype;

sum:real;

u:matrix;

myfile:text;

begin

assign(myfile,'holdmatrix4.txt');

rewrite(myfile);

{without using matrix}

writeln(mywork3, ' Check [f(p)]B');

writeln(' -------------');

writeln(mywork3);

writeln('Enter a ', n,' X ',n,' matrix on one line');

for j:= 1 to intpower(n,2) do

begin

read(x[j]);

end;

for j:= 1 to intpower(n,2) do

begin

sum:= 0;

for k:= 1 to intpower(n,2) do

begin

sum:= sum + coeff[j,k] * x[k];

end;

write(myfile,sum);

end;

reset(myfile);

for k:= 1 to intpower(n,2) do

begin

read(myfile,f[k,intpower(n,2) + 1]);

end;

close(myfile);

for j:= 1 to intpower(n,2) do

begin

for k:= 1 to intpower(n,2) + 1 do

begin

write(mywork3,f[j,k]:0:4,' ');

end;

writeln(mywork3);

end;

{Next Use operations on matrix f}

writeln(mywork3);

writeln(mywork3, 'Check [f(p)]B');

writeln(mywork3);

for l:= 1 to intpower(n,2) do

begin{l}

safeguard3(f,l,p,intpower(n,2),v);

if v = 0 then

begin{then}

operation13(f,l,intpower(n,2));

operation23(f,l,intpower(n,2));

end;{then}

end;{l}

writeln(mywork3);

solution(f,intpower(n,2));

{Now do using matrix M}

writeln(mywork3);

writeln(mywork3);

{To do}

for j:= 1 to intpower(n,2) do

begin

e[j,intpower(n,2) + 1]:= x[j];

end;

for j:= 1 to intpower(n,2) do

begin

for k:= 1 to intpower(n,2) + 1 do

begin

write(mywork3,e[j,k]:0:4,' ');

end;

writeln(mywork3);

end;

for l:= 1 to intpower(n,2) do

begin{l}

safeguard3(e,l,p,intpower(n,2),v);

if v = 0 then

begin{then}

operation13(e,l,intpower(n,2));

operation23(e,l,intpower(n,2));

end;{then}

end;{l}

writeln(mywork3);

solution(e,intpower(n,2));

writeln(mywork3);

writeln(mywork3);

for j:= 1 to intpower(n,2) do

begin

u[j,1]:= e[j,intpower(n,2) + 1];

end;

writeln(mywork3);

writeln(mywork3);

matrixmult(sol,u,prod);

end;

\(\)

begin

assign(mywork3,'mywork3.txt');

rewrite(mywork3);

getsize;

getcoefficients;

getmatrices;

hold(f,a,intpower(n,2)); {for me only-check [f(X)]}

hold(e,b,intpower(n,2));

{do determinants here}

{To do determinants to check bases here, then done.} {checking basis for A(2x2)}

PRODUCT:= 1;

for l:= 1 to intpower(n,2) do

begin{l}

safeguard2(b,l,p,intpower(n,2),v,PRODUCT);

if v = 0 then

begin{then}

operation11(b,l,intpower(n,2),PRODUCT);

operation21(b,l,intpower(n,2));

end;{then}

end;{l}

DETB:= DETERMINANT(b,intpower(n,2),PRODUCT);

writeln;

while detb = 0 do

begin

writeln('Set of matrices is not a basis for A ',n,' X ',n);

writeln;

getmatrices;

PRODUCT:= 1;

for l:= 1 to intpower(n,2) do

begin{l}

safeguard2(b,l,p,intpower(n,2),v,PRODUCT);

if v = 0 then

begin{then}

operation11(b,l,intpower(n,2),PRODUCT);

operation21(b,l,intpower(n,2));

end;{then}

end;{l}

DETB:= DETERMINANT(b,intpower(n,2),PRODUCT);

end;

{checking basis for Rn^2}

hold(c,a,intpower(n,2));

PRODUCT:= 1;

for l:= 1 to intpower(n,2) do

begin{l}

safeguard2(c,l,p,intpower(n,2),v,PRODUCT);

if v = 0 then

begin{then}

operation11(c,l,intpower(n,2),PRODUCT);

operation21(c,l,intpower(n,2));

end;{then}

end;{l}

DETA:= DETERMINANT(c,intpower(n,2),PRODUCT);

writeln;

while deta = 0 do

begin

writeln('Set of vectors is not a basis for R',intpower(n,2));

writeln;

getmatrices;

hold(c,a,intpower(n,2));

PRODUCT:= 1;

for l:= 1 to intpower(n,2) do

begin{l}

safeguard2(c,l,p,intpower(n,2),v,PRODUCT);

if v = 0 then

begin{then}

operation11(c,l,intpower(n,2),PRODUCT);

operation21(c,l,intpower(n,2));

end;{then}

end;{l}

DETA:= DETERMINANT(c,intpower(n,2),PRODUCT);

end;

{Only getting through if deta <> 0 and detb <> 0}

for l:= 1 to intpower(n,2) do

begin{l}

safeguard(a,l,p,v);

if v = 0 then

begin{then}

operation1(a,l);

operation2(a,l);

end;{then}

end;{l}

showdata;

check;

close(mywork3);

readln;

readln;

end.

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