A computer science problem by Rocco Dalto

Computer Science Level 4

Let f : R 3 → R 5 f: \mathbb R^3 \rightarrow \mathbb R^5 be linear transform defined by:

f ( x 1 x 2 x 3 ) = ( 3 ∗ x 1 − x 2 + x 3 x 2 + 4 ∗ x 3 5 ∗ x 1 − x 2 + x 3 4 ∗ x 1 − x 3 7 ∗ x 2 + x 3 ) f \left( \begin{array}{ccc} x_{1} \\ x_{2} \\ x_{3} \end{array} \right) = \left( \begin{array}{ccc} 3 * x_{1} - x_{2} + x_{3} \\ x_{2} + 4 * x_{3} \\ 5 * x_{1} - x_{2} + x_{3} \\ 4 * x{1} - x_{3} \\ 7 * x_{2} + x_{3} \end{array} \right)

and A = { ( 4 3 1 ) , ( 7 19 11 ) , ( 10 1 2 ) } A = \left \{ \left( \begin{array}{ccc} 4 \\ 3 \\ 1 \\ \end{array} \right), \left( \begin{array}{ccc} 7 \\ 19 \\ 11 \ \end{array} \right), \left( \begin{array}{ccc} 10 \\ 1 \\ 2 \ \end{array} \right) \right \} be a basis for R 3 . \mathbb R^3. .

and B = { ( 8 12 12 20 13 ) , ( 14 4 11 4 2 ) , ( 19 17 14 15 7 ) , ( 14 17 4 19 8 ) , ( 2 0 11 11 24 ) } B = \left \{ \left( \begin{array}{ccc} 8 \\ 12 \\ 12 \\ 20 \\ 13 \\ \end{array} \right), \left( \begin{array}{ccc} 14 \\ 4 \\ 11 \\ 4 \\ 2 \\ \end{array} \right), \left( \begin{array}{ccc} 19 \\ 17 \\ 14 \\ 15 \\ 7 \\ \end{array} \right), \left( \begin{array}{ccc} 14 \\ 17 \\ 4 \\ 19 \\ 8 \\ \end{array} \right), \left( \begin{array}{ccc} 2 \\ 0 \\ 11 \\ 11 \\ 24 \ \end{array} \right) \right \} be a basis for R 5 . \mathbb R^5. .

If M = [ a i j ] 5 x 3 M = [a_{ij}]_{5 \: x \: 3} represents the linear transform above find S = ∑ i = 1 5 ∑ j = 1 3 a i j \displaystyle S = \sum_{i = 1}^{5} \sum_{j = 1}^{3} a_{i j} . Express the result to 4 decimal places.

General Case:

Let f : R n → R m f: \mathbb R^n \rightarrow \mathbb R^m be linear transform defined by:

f ( x 1 x 2 . . . x n ) = ( c 11 ∗ x 1 + . . . + c 1 k ∗ x k + . . . + c 1 n ∗ x n c 21 ∗ x 1 + . . . + c 2 k ∗ x k + . . . + c 2 n ∗ x n . . . c j 1 ∗ x 1 + . . . + c j k ∗ x k + . . . + c j n ∗ x n . . . c m 1 ∗ x 1 + . . . + c m k ∗ x k + . . . + c m n ∗ x n ) f \left( \begin{array}{ccc} x_{1} \\ x_{2} \\ . \\ . \\ . \\ x_{n} \\ \end{array} \right) = \left( \begin{array}{ccc} c_{11} * x_{1} + ... + c_{1k} * x_{k}+ ... + c_{1n} * x_{n} \\ c_{21} * x_{1} + ... + c_{2k} * x_{k}+ ... + c_{2n} * x_{n} \\ . \\ . \\ . \\ c_{j1} * x_{1} + ... + c_{jk} * x_{k}+ ... + c_{jn} * x_{n} \\ . \\ . \\ . \\ c_{m1} * x_{1} + ... + c_{mk} * x_{k}+ ... + c_{mn} * x_{n} \end{array} \right)

Let V j = ( v 1 j v 2 j . . . v n j ) ∈ R n V_{j} = \left( \begin{array}{ccc} v_{1j} \\ v_{2j} \\ . \\ . \\ . \\ v_{nj} \\ \end{array} \right) \in \mathbb R^n

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

Let W j = ( w 1 j w 2 j . . . w m j ) ∈ R m W_{j} = \left( \begin{array}{ccc} w_{1j} \\ w_{2j} \\ . \\ . \\ . \\ w_{mj} \\ \end{array} \right) \in \mathbb R^m

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

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

Make certain A A and B B are bases for R n \mathbb R^n and R m \mathbb R^m .

You can use the program written to find the matrix M = [ a i j ] 5 x 3 M = [a_{ij}]_{5 \: x \: 3} that represents the linear transform above and output S = ∑ i = 1 5 ∑ j = 1 3 a i j \displaystyle S = \sum_{i = 1}^{5} \sum_{j = 1}^{3} a_{i j} .

Let X = ( 1 2 3 ) ∈ R 3 X = \left( \begin{array}{ccc} 1 \\ 2 \\ 3 \\ \end{array} \right) \in \mathbb R^3

To check the matrix M = [ a i j ] 5 x 3 M = [a_{ij}]_{5 \: x \: 3} found, first find [ f ( X ) ] B [f(X)]_B without using the matrix M = [ a i j ] 5 x 3 M = [a_{ij}]_{5 \: x \: 3} , then find [ f ( X ) ] B [f(X)]_B using the matrix M = [ a i j ] 5 x 3 M = [a_{ij}]_{5 \: x \: 3} .


The answer is 1.8571.

Rocco Dalto by Rocco Dalto , 67, USA

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1 solution

Rocco Dalto
Apr 11, 2017

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

program matrixrepresentaionoflineartransform;

{restricted to f:Rn --> Rm}

uses crt; 

                                {self contained for brillant}

const maxnum = 100;

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

 arraytype = array[1 .. maxnum] of real;

var coeff,b,a,sol,c:matrix; 

n,m,l,p,v:integer;

product,deta,detb:real;

mywork:text;

procedure getsize;

begin

writeln('Let f:Rn ---> Rm be linear transform ');

writeln('Enter n for Rn');

readln(n);

writeln('Enter m for Rm');

readln(m);

end;

procedure getcoefficients;

var j,k:integer;

begin

for j:= 1 to m do

begin

writeln('For row,', j, ', enter each coefficient of Xj for Rm');

for k:= 1 to n do

begin

read(coeff[j,k]);

end;

end;

end;

procedure getaugmentedmatrix;

var j,k,q,r,s:integer; 

vec:matrix;

sum:real;

myfile:text;

begin

assign(myfile,'holdmatrix.txt');

rewrite(myfile);

writeln('To Enter ', n, ' basis vectors for Rn: ');

for q:= 1 to n do

begin

write('Enter elements of vector ', q, ' : ');

for k:= 1 to n do

begin

read(b[k,q]);

end;

for j:= 1 to m do

begin

sum:= 0;

for k:= 1 to n do

begin

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

end;

vec[j,q]:= sum;

end;

end;

writeln('To Enter ', m, ' basis vectors for Rm: ');

for q:= 1 to m do

begin

write('Enter elements of vector ', q, ' : ');

for k:= 1 to m do

begin

read(a[k,q]);

end;

end;

for q:= 1 to m do

begin

for k:= 1 to m do

begin

write(myfile,a[q,k],' ');

end;

for r:= 1 to n do

begin

write(myfile,vec[q,r],' ');

end;

writeln(myfile);

end;

reset(myfile);

for j:= 1 to m do

begin

for k:= 1 to m + n do

begin

read(myfile,a[j,k]);

end;

end;

close(myfile);

{To show augmented matrix - for show work}

for q:= 1 to m do

begin

for k:= 1 to m do

begin

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

end;

for r:= 1 to n do

begin

write(mywork,vec[q,r]:0:0,' ');

end;

writeln(mywork);

end;

writeln(mywork);

writeln(mywork);

end;

procedure hold(var c:matrix; a:matrix);

var k,j:integer;

begin

for k:= 1 to m do

begin

for j:= 1 to m do

begin

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

end;

end;

end;

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}

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

var q:integer; 

z:real;

begin{switch}

for q:= 1 to m + n 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) <= n + m) 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 m + n do

begin{m}

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

end;{m}

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

for q:= 1 to m + n do

begin{m}

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

end;{m}

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

writeln(mywork);

for q:= 1 to m do

begin

for k:= 1 to m + n do

begin

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

end;

writeln(mywork);

end;

writeln(mywork);

end;{op1}

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

var q,r,k:integer; 

x:real;

s:matrix;

begin{op2}

for q:= 1 to m do

begin{q}

if q <> l then

begin{then}

x:= a[q,l];

for r:= 1 to m + n do

begin{m}

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

end;{m}

for r:= 1 to m + n do

begin{m}

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

end;{m}

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

end;{then}

end;{q}

writeln(mywork);

for q:= 1 to m do

begin

for k:= 1 to m + n do

begin

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

end;

writeln(mywork);

end;

writeln(mywork);

end;{op2}

PROCEDURE SHOWDATA;

VAR J,k:INTEGER; 

sum:extended;

myfile2:text;

BEGIN

assign(myfile2,'holdsolution.txt');

rewrite(myfile2);

writeln;

writeln('Matrix Representation of linear transformation f: Rn ---> Rm:');

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

writeln;

for j:= 1 to m do

begin

for k:= m + 1 to m + n do

begin

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

end;

writeln(myfile2);

end;

reset(myfile2);

for j:= 1 to m do

begin

for k:= 1 to n do

begin

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

end;

end;

close(myfile2);

for j:= 1 to m do

begin

for k:= 1 to n do

begin

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

end;

writeln;

end;

{For brillant only}

sum:= 0;

for j:= 1 to m do

begin

for k:= 1 to n do

begin

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

end;

end;

writeln;

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

END;

begin

assign(mywork,'mywork.txt');

rewrite(mywork);

getsize;

getcoefficients;

getaugmentedmatrix;

{add - to check each basis for Rn and Rm}

{checking basis for Rn}

PRODUCT:= 1;

for l:= 1 to n do

begin{l}

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

if v = 0 then

begin{then}

operation11(b,l,n,PRODUCT);

operation21(b,l,n);

end;{then}

end;{l}

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

writeln;

while detb = 0 do

begin

writeln('Set of vectors is not a basis for Rn');

writeln;

getaugmentedmatrix;

PRODUCT:= 1;

for l:= 1 to n do

begin{l}

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

if v = 0 then

begin{then}

operation11(b,l,n,PRODUCT);

operation21(b,l,n);

end;{then}

end;{l}

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

end;

{checking basis for Rm}

hold(c,a);

PRODUCT:= 1;

for l:= 1 to m do

begin{l}

safeguard2(c,l,p,m,v,PRODUCT);

if v = 0 then

begin{then}

operation11(c,l,m,PRODUCT);

operation21(c,l,m);

end;{then}

end;{l}

DETA:= DETERMINANT(c,m,PRODUCT);

writeln;

while deta = 0 do

begin

writeln('Set of vectors is not a basis for Rm');

writeln;

getaugmentedmatrix;

hold(c,a);

PRODUCT:= 1;

for l:= 1 to m do

begin{l}

safeguard2(c,l,p,m,v,PRODUCT);

if v = 0 then

begin{then}

operation11(c,l,m,PRODUCT);

operation21(c,l,m);

end;{then}

end;{l}

DETA:= DETERMINANT(c,m,PRODUCT);

end;

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

for l:= 1 to m 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;

readln;

readln;

end.

Running the above program we obtain:

Work shown in my file "mywork.txt":

[ 8 14 19 14 2 10 13 31 12 4 17 17 0 7 63 9 12 11 14 4 11 18 27 51 20 4 15 19 11 15 17 38 13 2 7 8 24 22 144 9 ] \left[ \begin{array}{ccccc|ccc} 8 & 14 & 19 & 14 & 2 & 10 & 13 & 31 \\ 12 & 4 & 17 & 17 & 0 & 7 & 63 & 9\\ 12 & 11 & 14 & 4 & 11 & 18 & 27 & 51\\ 20 & 4 & 15 & 19 & 11 & 15 & 17 & 38\\ 13 & 2 & 7 & 8 & 24 & 22 & 144 & 9\\ \ \end{array} \right]

where, f ( 4 3 1 ) = ( 10 7 18 15 22 ) f \left( \begin{array}{ccc} 4 \\ 3 \\ 1 \end{array} \right) = \left( \begin{array}{ccc} 10 \\ 7 \\ 18 \\ 15 \\ 22 \end{array} \right)

f ( 7 19 11 ) = ( 13 63 27 17 144 ) f \left( \begin{array}{ccc} 7 \\ 19 \\ 11 \end{array} \right) = \left( \begin{array}{ccc} 13 \\ 63 \\ 27 \\ 17 \\ 144 \end{array} \right)

f ( 10 1 2 ) = ( 31 9 51 38 9 ) f \left( \begin{array}{ccc} 10 \\ 1\\ 2 \end{array} \right) = \left( \begin{array}{ccc} 31 \\ 9 \\ 51 \\ 38\\ 9 \end{array} \right)

0.1250 ∗ R o w 1 0.1250 * Row1

[ 1.0000 1.7500 2.3750 1.7500 0.2500 1.2500 1.6250 3.8750 12.0000 4.0000 17.0000 17.0000 0.0000 7.0000 63.0000 9.0000 12.0000 11.0000 14.0000 4.0000 11.0000 18.0000 27.0000 51.0000 20.0000 4.0000 15.0000 19.0000 11.0000 15.0000 17.0000 38.0000 13.0000 2.0000 7.0000 8.0000 24.0000 22.0000 144.0000 9.0000 ] \left[ \begin{array}{ccccc|ccc} 1.0000 & 1.7500 & 2.3750 & 1.7500 & 0.2500 & 1.2500 & 1.6250 & 3.8750\\ 12.0000 & 4.0000 & 17.0000 & 17.0000 & 0.0000 & 7.0000 & 63.0000 & 9.0000\\ 12.0000 & 11.0000 & 14.0000 & 4.0000 & 11.0000 & 18.0000 & 27.0000 & 51.0000\\ 20.0000 & 4.0000 & 15.0000 & 19.0000 & 11.0000 & 15.0000 & 17.0000 & 38.0000\\ 13.0000 & 2.0000 & 7.0000 & 8.0000 & 24.0000 & 22.0000 & 144.0000 & 9.0000\\ \ \end{array} \right]

− 12.0000 ∗ R O W 1 + R O W 2 -12.0000 * ROW 1 + ROW2

− 12.0000 ∗ R O W 1 + R O W 3 -12.0000 * ROW 1 + ROW3

− 20.0000 ∗ R O W 1 + R O W 4 -20.0000 * ROW 1 + ROW4

− 13.0000 ∗ R O W 1 + R O W 5 -13.0000 * ROW 1 + ROW5

[ 1.0000 1.7500 2.3750 1.7500 0.2500 1.2500 1.6250 3.8750 0.0000 − 17.0000 − 11.5000 − 4.0000 − 3.0000 − 8.0000 43.5000 − 37.5000 0.0000 − 10.0000 − 14.5000 − 17.0000 8.0000 3.0000 7.5000 4.5000 0.0000 − 31.0000 − 32.5000 − 16.0000 6.0000 − 10.0000 − 15.5000 − 39.5000 0.0000 − 20.7500 − 23.8750 − 14.7500 20.7500 5.7500 122.8750 − 41.3750 ] \left[ \begin{array}{ccccc|ccc} 1.0000 & 1.7500 & 2.3750 & 1.7500 & 0.2500 & 1.2500 & 1.6250 & 3.8750\\ 0.0000 & -17.0000 & -11.5000 & -4.0000 & -3.0000 & -8.0000 & 43.5000 & -37.5000\\ 0.0000 & -10.0000 & -14.5000 & -17.0000 & 8.0000 & 3.0000 & 7.5000 & 4.5000\\ 0.0000 & -31.0000 & -32.5000 & -16.0000 & 6.0000 & -10.0000 & -15.5000 & -39.5000\\ 0.0000 & -20.7500 & -23.8750 & -14.7500 & 20.7500 & 5.7500 & 122.8750 & -41.3750\\ \ \end{array} \right]

− 0.0588 ∗ R o w 2 -0.0588 * Row2

[ 1.0000 1.7500 2.3750 1.7500 0.2500 1.2500 1.6250 3.8750 0.0000 1.0000 0.6765 0.2353 0.1765 0.4706 − 2.5588 2.2059 0.0000 − 10.0000 − 14.5000 − 17.0000 8.0000 3.0000 7.5000 4.5000 0.0000 − 31.0000 − 32.5000 − 16.0000 6.0000 − 10.0000 − 15.5000 − 39.5000 0.0000 − 20.7500 − 23.8750 − 14.7500 20.7500 5.7500 122.8750 − 41.3750 ] \left[ \begin{array}{ccccc|ccc} 1.0000 & 1.7500 & 2.3750 & 1.7500 & 0.2500 & 1.2500 & 1.6250 & 3.8750\\ 0.0000 & 1.0000 & 0.6765 & 0.2353 & 0.1765 & 0.4706 & -2.5588 & 2.2059\\ 0.0000 & -10.0000 & -14.5000 & -17.0000 & 8.0000 & 3.0000 & 7.5000 & 4.5000\\ 0.0000 & -31.0000 & -32.5000 & -16.0000 & 6.0000 & -10.0000 & -15.5000 & -39.5000\\ 0.0000 & -20.7500 & -23.8750 & -14.7500 & 20.7500 & 5.7500 & 122.8750 & -41.3750\\ \ \end{array} \right]

− 1.7500 ∗ R O W 2 + R O W 1 -1.7500 * ROW 2 + ROW1

10.0000 ∗ R O W 2 + R O W 3 10.0000 * ROW 2 + ROW3

31.0000 ∗ R O W 2 + R O W 4 31.0000 * ROW 2 + ROW4

20.7500 ∗ R O W 2 + R O W 5 20.7500 * ROW 2 + ROW5

[ 1.0000 0.0000 1.1912 1.3382 − 0.0588 0.4265 6.1029 0.0147 0.0000 1.0000 0.6765 0.2353 0.1765 0.4706 − 2.5588 2.2059 0.0000 0.0000 − 7.7353 − 14.6471 9.7647 7.7059 − 18.0882 26.5588 0.0000 0.0000 − 11.5294 − 8.7059 11.4706 4.5882 − 94.8235 28.8824 0.0000 0.0000 − 9.8382 − 9.8676 24.4118 15.5147 69.7794 4.3971 ] \left[ \begin{array}{ccccc|ccc} 1.0000 & 0.0000 & 1.1912 & 1.3382 & -0.0588 & 0.4265 & 6.1029 & 0.0147\\ 0.0000 & 1.0000 & 0.6765 & 0.2353 & 0.1765 & 0.4706 & -2.5588 & 2.2059\\ 0.0000 & 0.0000 & -7.7353 & -14.6471 & 9.7647 & 7.7059 & -18.0882 & 26.5588\\ 0.0000 & 0.0000 & -11.5294 & -8.7059 & 11.4706 & 4.5882 & -94.8235 & 28.8824\\ 0.0000 & 0.0000 & -9.8382 & -9.8676 & 24.4118 & 15.5147 & 69.7794 & 4.3971\\ \ \end{array} \right]

− 0.1293 ∗ R o w 3 -0.1293 * Row3

[ 1.0000 0.0000 1.1912 1.3382 − 0.0588 0.4265 6.1029 0.0147 0.0000 1.0000 0.6765 0.2353 0.1765 0.4706 − 2.5588 2.2059 0.0000 0.0000 1.0000 1.8935 − 1.2624 − 0.9962 2.3384 − 3.4335 0.0000 0.0000 − 11.5294 − 8.7059 11.4706 4.5882 − 94.8235 28.8824 0.0000 0.0000 − 9.8382 − 9.8676 24.4118 15.5147 69.7794 4.3971 ] \left[ \begin{array}{ccccc|ccc} 1.0000 & 0.0000 & 1.1912 & 1.3382 & -0.0588 & 0.4265 & 6.1029 & 0.0147\\ 0.0000 & 1.0000 & 0.6765 & 0.2353 & 0.1765 & 0.4706 & -2.5588 & 2.2059\\ 0.0000 & 0.0000 & 1.0000 & 1.8935 & -1.2624 & -0.9962 & 2.3384 & -3.4335\\ 0.0000 & 0.0000 & -11.5294 & -8.7059 & 11.4706 & 4.5882 & -94.8235 & 28.8824\\ 0.0000 & 0.0000 & -9.8382 & -9.8676 & 24.4118 & 15.5147 & 69.7794 & 4.3971\\ \ \end{array} \right]

− 1.1912 ∗ R O W 3 + R O W 1 -1.1912 * ROW 3 + ROW1

− 0.6765 ∗ R O W 3 + R O W 2 -0.6765 * ROW 3 + ROW2

11.5294 ∗ R O W 3 + R O W 4 11.5294 * ROW 3 + ROW4

9.8382 ∗ R O W 3 + R O W 5 9.8382 * ROW 3 + ROW5

[ 1.0000 0.0000 0.0000 − 0.9173 1.4449 1.6131 3.3175 4.1046 0.0000 1.0000 0.0000 − 1.0456 1.0304 1.1445 − 4.1407 4.5285 0.0000 0.0000 1.0000 1.8935 − 1.2624 − 0.9962 2.3384 − 3.4335 0.0000 0.0000 0.0000 13.1255 − 3.0837 − 6.8973 − 67.8631 − 10.7034 0.0000 0.0000 0.0000 8.7614 11.9924 5.7139 92.7852 − 29.3821 ] \left[ \begin{array}{ccccc|ccc} 1.0000 & 0.0000 & 0.0000 & -0.9173 & 1.4449 & 1.6131 & 3.3175 & 4.1046\\ 0.0000 & 1.0000 & 0.0000 & -1.0456 & 1.0304 & 1.1445 & -4.1407 & 4.5285\\ 0.0000 & 0.0000 & 1.0000 & 1.8935 & -1.2624 & -0.9962 & 2.3384 & -3.4335\\ 0.0000 & 0.0000 & 0.0000 & 13.1255 & -3.0837 & -6.8973 & -67.8631 & -10.7034\\ 0.0000 & 0.0000 & 0.0000 & 8.7614 & 11.9924 & 5.7139 & 92.7852 & -29.3821\\ \ \end{array} \right]

0.0762 ∗ R o w 4 0.0762 * Row4

[ 1.0000 0.0000 0.0000 − 0.9173 1.4449 1.6131 3.3175 4.1046 0.0000 1.0000 0.0000 − 1.0456 1.0304 1.1445 − 4.1407 4.5285 0.0000 0.0000 1.0000 1.8935 − 1.2624 − 0.9962 2.3384 − 3.4335 0.0000 0.0000 0.0000 1.0000 − 0.2349 − 0.5255 − 5.1703 − 0.8155 0.0000 0.0000 0.0000 8.7614 11.9924 5.7139 92.7852 − 29.3821 ] \left[ \begin{array}{ccccc|ccc} 1.0000 & 0.0000 & 0.0000 & -0.9173 & 1.4449 & 1.6131 & 3.3175 & 4.1046 \\ 0.0000 & 1.0000 & 0.0000 & -1.0456 & 1.0304 & 1.1445 & -4.1407 & 4.5285\\ 0.0000 & 0.0000 & 1.0000 & 1.8935 & -1.2624 & -0.9962 & 2.3384 & -3.4335\\ 0.0000 & 0.0000 & 0.0000 & 1.0000 & -0.2349 & -0.5255 & -5.1703 & -0.8155\\ 0.0000 & 0.0000 & 0.0000 & 8.7614 & 11.9924 & 5.7139 & 92.7852 & -29.3821\\ \ \end{array} \right]

0.9173 ∗ R O W 4 + R O W 1 0.9173 * ROW 4 + ROW1

1.0456 ∗ R O W 4 + R O W 2 1.0456 * ROW 4 + ROW2

− 1.8935 ∗ R O W 4 + R O W 3 -1.8935 * ROW 4 + ROW3

− 8.7614 ∗ R O W 4 + R O W 5 -8.7614 * ROW 4 + ROW5

[ 1.0000 0.0000 0.0000 0.0000 1.2294 1.1311 − 1.4253 3.3565 0.0000 1.0000 0.0000 0.0000 0.7848 0.5950 − 9.5469 3.6758 0.0000 0.0000 1.0000 0.0000 − 0.8175 − 0.0012 12.1286 − 1.8893 0.0000 0.0000 0.0000 1.0000 − 0.2349 − 0.5255 − 5.1703 − 0.8155 0.0000 0.0000 0.0000 0.0000 14.0508 10.3179 138.0846 − 22.2375 ] \left[ \begin{array}{ccccc|ccc} 1.0000 & 0.0000 & 0.0000 & 0.0000 & 1.2294 & 1.1311 & -1.4253 & 3.3565\\ 0.0000 & 1.0000 & 0.0000 & 0.0000 & 0.7848 & 0.5950 & -9.5469 & 3.6758\\ 0.0000 & 0.0000 & 1.0000 & 0.0000 & -0.8175 & -0.0012 & 12.1286 & -1.8893\\ 0.0000 & 0.0000 & 0.0000 & 1.0000 & -0.2349 & -0.5255 & -5.1703 & -0.8155\\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 14.0508 & 10.3179 & 138.0846 & -22.2375\\ \ \end{array} \right]

0.0712 ∗ R o w 5 0.0712 * Row5

[ 1.0000 0.0000 0.0000 0.0000 1.2294 1.1311 − 1.4253 3.3565 0.0000 1.0000 0.0000 0.0000 0.7848 0.5950 − 9.5469 3.6758 0.0000 0.0000 1.0000 0.0000 − 0.8175 − 0.0012 12.1286 − 1.8893 0.0000 0.0000 0.0000 1.0000 − 0.2349 − 0.5255 − 5.1703 − 0.8155 0.0000 0.0000 0.0000 0.0000 1.0000 0.7343 9.8275 − 1.5827 ] \left[ \begin{array}{ccccc|ccc} 1.0000 & 0.0000 & 0.0000 & 0.0000 & 1.2294 & 1.1311 & -1.4253 & 3.3565\\ 0.0000 & 1.0000 & 0.0000 & 0.0000 & 0.7848 & 0.5950 & -9.5469 & 3.6758\\ 0.0000 & 0.0000 & 1.0000 & 0.0000 & -0.8175 & -0.0012 & 12.1286 & -1.8893\\ 0.0000 & 0.0000 & 0.0000 & 1.0000 & -0.2349 & -0.5255 & -5.1703 & -0.8155\\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.0000 & 0.7343 & 9.8275 & -1.5827\\ \ \end{array} \right]

− 1.2294 ∗ R O W 5 + R O W 1 -1.2294 * ROW 5 + ROW1

− 0.7848 ∗ R O W 5 + R O W 2 -0.7848 * ROW 5 + ROW2

0.8175 ∗ R O W 5 + R O W 3 0.8175 * ROW 5 + ROW3

0.2349 ∗ R O W 5 + R O W 4 0.2349 * ROW 5 + ROW4

[ 1.0000 0.0000 0.0000 0.0000 0.0000 0.2283 − 13.5069 5.3022 0.0000 1.0000 0.0000 0.0000 0.0000 0.0187 − 17.2592 4.9178 0.0000 0.0000 1.0000 0.0000 0.0000 0.5992 20.1626 − 3.1832 0.0000 0.0000 0.0000 1.0000 0.0000 − 0.3530 − 2.8615 − 1.1873 0.0000 0.0000 0.0000 0.0000 1.0000 0.7343 9.8275 − 1.5827 ] \left[ \begin{array}{ccccc|ccc} 1.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.2283 & -13.5069 & 5.3022\\ 0.0000 & 1.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0187 & -17.2592 & 4.9178\\ 0.0000 & 0.0000 & 1.0000 & 0.0000 & 0.0000 & 0.5992 & 20.1626 & -3.1832\\ 0.0000 & 0.0000 & 0.0000 & 1.0000 & 0.0000 & -0.3530 & -2.8615 & -1.1873\\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.0000 & 0.7343 & 9.8275 & -1.5827\\ \ \end{array} \right]

M = [ 0.2283 − 13.5069 5.3022 0.0187 − 17.2592 4.9178 0.5992 20.1626 − 3.1832 − 0.3530 − 2.8615 − 1.1873 0.7343 9.8275 − 1.5827 ] M = \begin{bmatrix}{0.2283} && {-13.5069} && {5.3022} \\ {0.0187} && {-17.2592} && {4.9178} \\ {0.5992} && {20.1626} && {-3.1832} \\ {-0.3530} && {-2.8615} && {-1.1873} \\ {0.7343} && {9.8275} && {-1.5827}\end{bmatrix}

and ∑ i = 1 5 ∑ j = 1 3 a i j = 1.8571 \sum_{i = 1}^{5} \sum_{j = 1}^{3} a_{i j} = \boxed{1.8571} .

I extended the program(extension not shown above) to check matrix M M above.

Let X = [ 1 2 3 ] ∈ R 3 X = \left[ \begin{array}{ccc} 1 \\ 2 \\ 3 \\ \end{array} \right] \in \mathbb R^3

Without using matrix M M :

f ( 1 2 3 ) = [ 4 14 6 1 17 ] f \left( \begin{array}{ccc} 1\\ 2\\ 3\end{array} \right) = \left[ \begin{array}{ccc} 4 \\ 14 \\ 6 \\ 1\\ 17 \end{array} \right]

Using the set of basis vectors for B B and f ( X ) f(X) we set up the system:

[ 8.0000 14.0000 19.0000 14.0000 2.0000 4.0000 12.0000 4.0000 17.0000 17.0000 0.0000 14.0000 12.0000 11.0000 14.0000 4.0000 11.0000 6.0000 20.0000 4.0000 15.0000 19.0000 11.0000 1.0000 13.0000 2.0000 7.0000 8.0000 24.0000 17.0000 ] \left[ \begin{array}{ccccc|c} 8.0000 & 14.0000 & 19.0000 & 14.0000 & 2.0000 & 4.0000 \\ 12.0000 & 4.0000 & 17.0000 & 17.0000 & 0.0000 & 14.0000 \\ 12.0000 & 11.0000 & 14.0000 & 4.0000 & 11.0000 & 6.0000 \\ 20.0000 & 4.0000 & 15.0000 & 19.0000 & 11.0000 & 1.0000 \\ 13.0000 & 2.0000 & 7.0000 & 8.0000 & 24.0000 & 17.0000 \\ \ \end{array} \right]

Solving the system we obtain:

[ f ( X ) ] B = [ − 2.1155 − 3.1958 4.0341 − 0.9654 1.2657 ] [f(X)]_B = \left[ \begin{array}{ccccc} -2.1155 \\ -3.1958 \\ 4.0341 \\ -0.9654 \\ 1.2657 \\ \end{array} \right]

Using matrix M M :

Using the set of basis vectors for A A and X X we set up the system:

[ 4.0000 7.0000 10.0000 1.0000 3.0000 19.0000 1.0000 2.0000 1.0000 11.0000 2.0000 3.0000 ] \left[ \begin{array}{ccc|c} 4.0000 & 7.0000 & 10.0000 & 1.0000 \\ 3.0000 & 19.0000 & 1.0000 & 2.0000 \\ 1.0000 & 11.0000 & 2.0000 & 3.0000 \\ \end{array} \right]

Solving the system we obtain:

[ X ] A = [ − 1.5493 0.3239 0.4930 ] [X]_A = \left[ \begin{array}{ccccc} -1.5493 \\ 0.3239 \\ 0.4930\\ \end{array} \right]

[ f ( X ) ] B = M ∗ [ X ] A = [ 0.2283 − 13.5069 5.3022 0.0187 − 17.2592 4.9178 0.5992 20.1626 − 3.1832 − 0.3530 − 2.8615 − 1.1873 0.7343 9.8275 − 1.5827 ] ∗ [ − 1.5493 0.3239 0.4930 ] = [ − 2.1155 − 3.1958 4.0341 − 0.9654 1.2657 ] [f(X)]_B = M * [X]_A = \begin{bmatrix}{0.2283} && {-13.5069} && {5.3022} \\ {0.0187} && {-17.2592} && {4.9178} \\ {0.5992} && {20.1626} && {-3.1832} \\ {-0.3530} && {-2.8615} && {-1.1873} \\ {0.7343} && {9.8275} && {-1.5827}\end{bmatrix} * \left[ \begin{array}{ccccc} -1.5493 \\ 0.3239 \\ 0.4930\\ \end{array} \right] = \left[ \begin{array}{ccccc} -2.1155 \\ -3.1958 \\ 4.0341 \\ -0.9654 \\ 1.2657 \\ \end{array} \right]

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