An algebra problem by Rocco Dalto

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

$$

Since $f: \mathbb C^2 \rightarrow \mathbb C^3$ is a linear transform, for each integer $j \ni (1 <= j <= 2)$

$$

$f \left( \begin{array}{ccc} z_{1j} \\ z_{2j} \\ \end{array} \right) = \alpha_{1j} * \left( \begin{array}{ccc} 1 + 2i\\ 3 + 4i \\ 5 + 6i \\ \end{array} \right) + \alpha_{2j} * \left( \begin{array}{ccc} -7 + 10i \\ 2 - 6i \\ 1 + 2i\ \end{array} \right) + \alpha_{3j} * \left( \begin{array}{ccc} -6 + 5i \\ 10 + 3i \\ 5 + i \ \end{array} \right)$

$= \begin{bmatrix}{1 + 2i} && {-7 + 10i} && {-6 + 5i} \\ {3 + 4i} && {2 - 6i} && {10 + 3i} \\ {5 + 6i} && {1 + 2i} && {5 + i} \end{bmatrix} * \begin{vmatrix}{\alpha_{1j}} \\{\alpha_{2j}} \\ {\alpha_{3j}} \end{vmatrix}$

where $\alpha_{1j}, \alpha_{2j}, \alpha_{3j} \in \mathbb C$ and $f \left( \begin{array}{ccc} 2 - 5i \\ 7 + 3i \\ \end{array} \right) = \left( \begin{array}{ccc} 25 - 28i\\ 15 + 6i\\ 29 - 35i\\ \end{array} \right)$

$f \left( \begin{array}{ccc} 10 - 6i \\ 3 + 4i \\ \end{array} \right) = \left( \begin{array}{ccc} 23 - 26i\\ 13 - 32i\\ 96 - 74i\\ \end{array} \right)$

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

two 3 X 3 systems of equations.

$$

$\left[ \begin{array}{ccc|cc} 1+ 2i & -7 + 10i & -6 + 5i & 25 - 28i & 23 - 26i \\ 3 + 4i & 2 - 6i & 10 + 3i & 15 + 6i & 13 - 32i \\ 5 + 6i & 1 + 2i & 5 + i & 29 - 35i & 96 - 74i \\ \ \end{array} \right]$

$$

$(0.2000 - 0.4000i) * ROW_{1}$

$$

$\left[ \begin{array}{ccc|cc} 1.0000 + 0.0000i & 2.6000 + 4.8000i & 0.8000 + 3.4000i & -6.2000 - 15.6000i & -5.8000 - 14.4000i \\ 3.0000 + 4.0000i & 2.0000 - 6.0000i & 10.0000 + 3.0000i & 15.0000 + 6.0000i & 13.0000 - 32.0000i \\ 5.0000 + 6.0000i & 1.0000 + 2.0000i & 5.0000 + 1.0000i & 29.0000 - 35.0000i & 96.0000 - 74.0000i \\ \ \end{array} \right]$

$$

$(-3.0000 - 4.0000i ) * ROW_{1} + ROW_{2}$

$(-5.0000 - 6.0000i ) * ROW_{1} + ROW_{3}$

$$

$\left[ \begin{array}{ccc|cc} 1.0000 + 0.0000i & 2.6000 + 4.8000i & 0.8000 + 3.4000i & -6.2000 - 15.6000i & -5.8000 - 14.4000i \\ 0.0000 + 0.0000i & 13.4000 - 30.8000i & 21.2000 -10.4000i & -28.8000 + 77.6000i & -27.2000 + 34.4000i \\ 0.0000 + i 0.0000 & 16.8000 - 37.6000i & 21.4000 - 20.8000i & -33.6000 + 80.2000i & 38.6000 + 32.8000i \\ \ \end{array} \right]$

$$

$(0.0119 + 0.0273i) * ROW_{2}$

$$

$\left[ \begin{array}{ccc|cc} 1.0000 + 0.0000i & 2.6000 + 4.8000i & 0.8000 + 3.4000i & -6.2000 - 15.6000i & -5.8000 - 14.4000i \\ 0.0000 + 0.0000i & 1.0000 + 0.0000i & 0.5357 + 0.4552i & -2.4606 + 0.1354i & -1.2622 - 0.3340i \\ 0.0000 + 0.0000i & 16.8000 - 37.6000i & 21.4000 - 20.8000i & -33.6000 + 80.2000i & 38.6000 + i32.8000i \\ \ \end{array} \right]$

$$

$(-2.6000 - 4.8000i ) * ROW_{2} + ROW_{1}$

$(-16.8000 + 37.6000i ) * ROW_{2} + ROW_{3}$

$$

$\left[ \begin{array}{ccc|cc} 1.0000 + 0.0000i & 0.0000 + 0.0000i & 1.5923 - 0.3551i & 0.8475 - 4.1415i & -4.1214 - 7.4731i \\ 0.0000 + 0.0000i & 1.0000 + 0.0000i & 0.5357 + 0.4552i & -2.4606 + 0.1354i & -1.2622 - 0.3340i \\ 0.0000 + 0.0000i & 0.0000 + 0.0000i & -4.7171 - 8.3049i & 2.6449 - 14.5923i & 72.3625 - 9.0473i \\ \ \end{array} \right]$

$$

$(-0.0517 + 0.0910i) * ROW_{3}$

$$

$\left[ \begin{array}{ccc|cc} 1.0000 + 0.0000i & 0.0000 + 0.0000i & 1.5923 - 0.3551i & 0.8475 - 4.1415i & -4.1214 - 7.4731i \\ 0.0000 + 0.0000i & 1.0000 + 0.0000i & 0.5357 + 0.4552i & -2.4606 + 0.1354i & -1.2622 - 0.3340i \\ 0.0000 + 0.0000i & 0.0000 + 0.0000i & 1.0000 + 0.0000i & 1.1917 + 0.9954i & -2.9182 + 7.0557i \\ \ \end{array} \right]$

$$

$(-1.5923 + 0.3551i ) * ROW_{3} + ROW_{1}$

$(-0.5357 - 0.4552i ) * ROW_{3} + ROW_{2}$

$$

$\left[ \begin{array}{ccc|cc} 1.0000 + 0.0000i & 0.0000 + 0.0000i & 0.0000 + 0.0000i & -1.4034 - 5.3032i & -1.9803 - 19.7440i \\ 0.0000 + 0.0000i & 1.0000 + 0.0000i & 0.0000 + 0.0000i & -2.6459 - 0.9403i & 3.5132 - 2.7854i \\ 0.0000 + 0.0000i & 0.0000 + 0.0000i & 1.0000 + 0.0000i & 1.1917 + 0.9954i & -2.9182 + 7.0557i \\ \ \end{array} \right]$

$\implies$

$$

$M = \begin{bmatrix}{-1.4034 - 5.3032i } && {-1.9803 - 19.7440i} \\ {-2.6459 - 0.9403i} && {3.5132 - 2.7854i }\\ {1.1917 + 0.9954i} && {-2.9182 + 7.0557i } \end{bmatrix}$

$$

and,

$\displaystyle \sum_{j = 1}^{3} \sum_{k = 1}^{2} (a_{jk} + b_{jk}i) = -1.7265 + 8.0511i = \alpha + \beta i$

$\implies$

$\| \alpha + \beta i \| = \sqrt{\alpha^2 + \beta^2} = \boxed{8.2341}$ .

$$

...

I wrote the program in Free Pascal.

$$

program matrixrepresentaionoflineartransform_complex;

{restricted to f:Cn --> Cm}

uses crt;

                                {self contained for brillant}

const maxnum = 100;

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

var coeffa,coeffb,reb,imgb,rea,imga,resol,imgsol:matrix;

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

mywork:text;

$$

function power(base:real; exponent:integer):real;

var n:integer;

product:real;

begin

product:= 1;

for n:= 1 to exponent do

product:= product * base;

power:= product;

end;

$$

procedure getsize;

begin

writeln('Let f:Cn ---> Cm be linear transform ');

writeln('Enter n for Cn');

readln(n);

writeln('Enter m for Cm');

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 Zj for Cm');

for k:= 1 to n do

begin

read(coeffa[j,k]);

read(coeffb[j,k]);

end;

end;

end;

$$

procedure complexinverse(var reinvz,imginvz:real; rez,imgz:real);

var lengthsqr:real;

begin

lengthsqr:= power(rez,2) + power(imgz,2);

reinvz:= rez/lengthsqr;

imginvz:= -imgz/lengthsqr;

end;

$$

function reproduct(rez1,imgz1,rez2,imgz2:real):real;

begin

reproduct:= rez1 * rez2 - imgz1 * imgz2;

end;

$$

function imgproduct(rez1,imgz1,rez2,imgz2:real):real;

begin

imgproduct:= rez2 * imgz1 + rez1 * imgz2;

end;

$$

procedure getaugmentedmatrix;

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

revec,imgvec:matrix;

resum,imgsum:real;

myfile:text;

begin

assign(myfile,'holdcmatrix.txt');

rewrite(myfile);

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

for q:= 1 to n do

begin

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

for k:= 1 to n do

begin

read(reb[k,q]);

read(imgb[k,q]);

end;

for j:= 1 to m do

begin

resum:= 0;

imgsum:= 0;

for k:= 1 to n do

begin

resum:= resum + reproduct(coeffa[j,k],coeffb[j,k],reb[k,q],imgb[k,q]);

imgsum:= imgsum + imgproduct(coeffa[j,k],coeffb[j,k],reb[k,q],imgb[k,q]);

end;

revec[j,q]:= resum;

imgvec[j,q]:= imgsum;

end;

end;

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

for q:= 1 to m do

begin

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

for k:= 1 to m do

begin

read(rea[k,q]);

read(imga[k,q]);

end;

end;

for q:= 1 to m do

begin

for k:= 1 to m do

begin

write(myfile,rea[q,k]:0:4,' ',imga[q,k],' ');

end;

for r:= 1 to n do

begin

write(myfile,revec[q,r]:0:4,' ',imgvec[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,rea[j,k]);

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

end;

end;

close(myfile);

{To show augmented matrix - for show work}

writeln(mywork,' Augmented Matrix');

writeln(mywork);

for q:= 1 to m do

begin

for k:= 1 to m do

begin

write(mywork,rea[q,k]:0:4,' + i', imga[q,k]:0:4,' ');

end;

for r:= 1 to n do

begin

write(mywork,revec[q,r]:0:4,' + i', imgvec[q,r]:0:4,' ');

end;

writeln(mywork);

end;

writeln(mywork);

writeln(mywork);

end;

$$

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

var q:integer;

z,w: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;

   w:= d[l,q];

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

   d[p,q]:= w;

end;{m}

end;{switch}

$$

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

var q:integer;

begin{safeguard}

v:=0;

if (c[l,l] = 0) and (d[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) or (d[q + 1,l] <> 0) then

begin{then}

        p:= q + 1;

        switch(c,d,l,p);

        v:= 0;

end;{then}

q:= q + 1;

end;{loop}

end;{then}

end;{safeguard}

$$

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

var q,k:integer;

res,imgs:matrix;

reinvz,imginvz:real;

begin{op1}

for q:= 1 to m + n do

begin{q}

complexinverse(reinvz,imginvz,a[l,l],b[l,l]);

res[l,q]:= reproduct(a[l,q],b[l,q],reinvz,imginvz);

imgs[l,q]:= imgproduct(a[l,q],b[l,q],reinvz,imginvz);

end;{q}

for q:= 1 to m + n do

begin{q}

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

b[l,q]:= imgs[l,q];

end;{q}

writeln(mywork,'(',reinvz:0:4,' + i', imginvz:0:4,') * ROW ', l);

writeln(mywork);

writeln(mywork);

for q:= 1 to m do

begin

for k:= 1 to m + n do

begin

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

end;

writeln(mywork);

end;

writeln(mywork);

writeln(mywork);

end;{op1}

$$

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

var q,r,k:integer;

x,y:real;

res,imgs:matrix;

begin{op2}

for q:= 1 to m do

begin{q}

if q <> l then

begin{then}

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

y:= -1 * b[q,l];

for r:= 1 to m + n do

begin{r}

res[q,r]:= reproduct(x,y,a[l,r],b[l,r]) + a[q,r];

imgs[q,r]:= imgproduct(x,y,a[l,r],b[l,r]) + b[q,r];

end;{r}

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

for r:= 1 to m + n do

begin{r}

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

b[q,r]:= imgs[q,r];

end;{r}

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,' + i ',b[q,k]:0:4,' ');

end;

writeln(mywork);

end;

writeln(mywork);

writeln(mywork);

end;{op2}

$$

PROCEDURE SHOWDATA;

VAR J,k:INTEGER;

resum,imgsum:extended;

myfile2:text;

BEGIN

assign(myfile2,'holdsolution6.txt');

rewrite(myfile2);

writeln;

writeln('Matrix Representation of linear transformation f: Cn ---> Cm:');

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

writeln;

for j:= 1 to m do

begin

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

begin

write(myfile2,rea[j,k]:0:4,' ',imga[j,k],' ');

end;

writeln(myfile2);

end;

reset(myfile2);

for j:= 1 to m do

begin

for k:= 1 to n do

begin read(myfile2,resol[j,k]);

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

end;

end;

close(myfile2);

for j:= 1 to m do

begin

for k:= 1 to n do

begin

write(resol[j,k]:0:4,' + i', imgsol[j,k]:0:4,' ');

end;

writeln;

end;

{For brillant only.}

for j:= 1 to m do

begin

resum:= 0;

imgsum:= 0;

for k:= 1 to n do

begin

resum:= resum + resol[j,k];

imgsum:= imgsum + imgsol[j,k];

end;

end;

writeln(mywork);

writeln(mywork);

writeln(mywork, 'Sum = ',resum:0:4,' + i', imgsum:0:4);

writeln(mywork, '|Sum| = ',sqrt(power(resum,2) + power(imgsum,2)):0:4);

writeln;

writeln;

writeln('Sum = ',resum:0:4,' + i', imgsum:0:4);

writeln('|Sum| = ',sqrt(power(resum,2) + power(imgsum,2)):0:4);

END;

$$

begin

assign(mywork,'cmywork.txt');

rewrite(mywork);

getsize;

getcoefficients;

getaugmentedmatrix;

for l:= 1 to m do

begin{l}

safeguard(rea,imga,l,p,v);

if v = 0 then

begin{then}

operation1(rea,imga,l);

operation2(rea,imga,l);

end;{then}

end;{l}

showdata;

close(mywork);

readln;

readln;

end.