Chapter 4, Problem 1E

Use MATLAB to generate a matrix W and a vector x by setting
$W=triu(ones(5))$
and
$x={[1:5]}^{\text{'}}$
The columns of W can be used to form an ordered basis
$F=\{w_1,w_2,w_3,w_4,w_5\}$
Let $L:{ℝ}^5\to {ℝ}^5$ be a linear operator such that
$\begin{array}{ccc}L(w_1)=w_2,& L(w_2)=w_3,& L(w_3)=w_4\end{array}$
and
$L(w_4)=4w_1+3w_2+2w_3+w_4$
$L(w_5)=w_1+w_2+w_3+3w_4+w_5$
(a) Determine the matrix A representing L with respect to F, and enter it in MATLAB.
(b) Use MATLAB to compute the coordinate vector $y=W^{-1}x$ of x with respect to F.
(c) Use A to compute the coordinate vector z of $L(x)$ with respect to F.
(d) W is the transition matrix from F to the standard basis for ${ℝ}^5$ . Use W to compute the coordinate vector of $L(x)$ with respect to the standard basis.

To determine

Calculate the matrix A using given relation.

Answer to Problem 1E

The solution is

  $A=\left[\begin{array}{l}\text{ 0 0 5 1}\\ \text{ 1 0 4 2 }\\ \text{ 0 1 3 3}\\ \text{ 0 0 2 4}\end{array}\right]$

Explanation of Solution

Given:The information has been given

  $\begin{array}{l}W=triu(ones(4))\\ x=[1:4]\text{'}\\ F=\left\{\begin{array}{cccc}{w}_1,& w_2,& w_3,& w_4\end{array}\right\}\\ L(w_1)=w_2\\ L(w_2)=w_3\\ L(w_3)=w_4\\ L(w_4)=4w_1+3w_2+2w_3+w_4\end{array}$

Concept Used:

Given,

  $\begin{array}{l}W=triu(ones(4))\\ x=[1:4]\text{'}\\ F=\left\{\begin{array}{cccc}{w}_1,& w_2,& w_3,& w_4\end{array}\right\}\\ L(w_1)=w_2\\ L(w_2)=w_3\\ L(w_3)=w_4\\ L(w_4)=4w_1+3w_2+2w_3+w_4\end{array}$

Using the above relation, we will calculate the matrix A representing L with respect to F.

Program:

clc

clear

close all

W = triu(ones(4));

x = (1:4)';

A = [0 0 5 1;

1 0 4 2;

0 1 3 3;

0 0 2 4];

y = W^-1*x;

z = A*y;

Lx = W*z;

fprintf('The matrix A is: \n')

disp(A)

fprintf('The relation W^-1*x is: \n')

disp(y)

fprintf('The coordinate vector z is: \n')

disp(z)

fprintf('The coordinate vector of Lx is: \n')

disp(Lx)

Quarry:

• First, we have defined the given matrix W.

Then define the matrix x.

Calculate the matrix A and write.

To determine

Calculate the given relation using relations.

Answer to Problem 1E

The solution is

  $y=W^{-1}x=\left[\begin{array}{l}\text{-1}\\ \text{-1}\\ \text{-1}\\ \text{ 4}\end{array}\right]$

Explanation of Solution

Given:The information has been given

  $\begin{array}{l}W=triu(ones(4))\\ x=[1:4]\text{'}\\ F=\left\{\begin{array}{cccc}{w}_1,& w_2,& w_3,& w_4\end{array}\right\}\\ L(w_1)=w_2\\ L(w_2)=w_3\\ L(w_3)=w_4\\ L(w_4)=4w_1+3w_2+2w_3+w_4\end{array}$

Concept Used:

Given,

  $\begin{array}{l}W=triu(ones(4))\\ x=[1:4]\text{'}\\ F=\left\{\begin{array}{cccc}{w}_1,& w_2,& w_3,& w_4\end{array}\right\}\\ L(w_1)=w_2\\ L(w_2)=w_3\\ L(w_3)=w_4\\ L(w_4)=4w_1+3w_2+2w_3+w_4\end{array}$

Calculate $y=W^{-1}x$

After calculating, we will get

  $y=W^{-1}x=\left[\begin{array}{l}\text{-1}\\ \text{-1}\\ \text{-1}\\ \text{ 4}\end{array}\right]$

Program:

clc
clear
close all
W = triu(ones(4));
x = (1:4)';
A = [0 0 5 1;
     1 0 4 2;
     0 1 3 3;
     0 0 2 4]; 
y = W^-1*x; 
z = A*y; 
Lx = W*z;
fprintf('The matrix A is: \n')
disp(A)
fprintf('The relation W^-1*x is: \n')
disp(y)
fprintf('The coordinate vector z is: \n')
disp(z)
fprintf('The coordinate vector of Lx is: \n')
disp(Lx)

Quarry:

• First, we have defined the given matrix W.
• Then define the matrix x.
• Calculate the relation $y=W^{-1}x$ .

To determine

Using the matrix A calculate the coordinate vector.

Answer to Problem 1E

The solution is

  $z=\left[\begin{array}{l}\text{-1}\\ \text{ 3}\\ \text{ 8}\\ \text{14}\end{array}\right]$

Explanation of Solution

Given:The information has been given

  $\begin{array}{l}W=triu(ones(4))\\ x=[1:4]\text{'}\\ F=\left\{\begin{array}{cccc}{w}_1,& w_2,& w_3,& w_4\end{array}\right\}\\ L(w_1)=w_2\\ L(w_2)=w_3\\ L(w_3)=w_4\\ L(w_4)=4w_1+3w_2+2w_3+w_4\end{array}$

Concept Used:

Given,

  $\begin{array}{l}W=triu(ones(4))\\ x=[1:4]\text{'}\\ F=\left\{\begin{array}{cccc}{w}_1,& w_2,& w_3,& w_4\end{array}\right\}\\ L(w_1)=w_2\\ L(w_2)=w_3\\ L(w_3)=w_4\\ L(w_4)=4w_1+3w_2+2w_3+w_4\end{array}$

Calculate coordinate vector $z=A{W}^{-1}X=Ay$ .

After calculating, we will get

  $z=\left[\begin{array}{l}\text{-1}\\ \text{ 3}\\ \text{ 8}\\ \text{14}\end{array}\right]$

Program:

clc
clear
close all
W = triu(ones(4));
x = (1:4)';
A = [0 0 5 1;
     1 0 4 2;
     0 1 3 3;
     0 0 2 4]; 
y = W^-1*x; 
z = A*y; 
Lx = W*z;
fprintf('The matrix A is: \n')
disp(A)
fprintf('The relation W^-1*x is: \n')
disp(y)
fprintf('The coordinate vector z is: \n')
disp(z)
fprintf('The coordinate vector of Lx is: \n')
disp(Lx)

Quarry:

• First, we have defined the given matrix W.
• Then define the matrix x.
• Calculate coordinate vector z.

To determine

Calculate the coordinate vector of Lx with respect to standard basis.

Answer to Problem 1E

The solution is

  $Lx=\left[\begin{array}{l}\text{24}\\ \text{25}\\ \text{22}\\ \text{14}\end{array}\right]$

Explanation of Solution

Given:The information has been given

  $\begin{array}{l}W=triu(ones(4))\\ x=[1:4]\text{'}\\ F=\left\{\begin{array}{cccc}{w}_1,& w_2,& w_3,& w_4\end{array}\right\}\\ L(w_1)=w_2\\ L(w_2)=w_3\\ L(w_3)=w_4\\ L(w_4)=4w_1+3w_2+2w_3+w_4\end{array}$

Concept Used:

Given,

  $\begin{array}{l}W=triu(ones(4))\\ x=[1:4]\text{'}\\ F=\left\{\begin{array}{cccc}{w}_1,& w_2,& w_3,& w_4\end{array}\right\}\\ L(w_1)=w_2\\ L(w_2)=w_3\\ L(w_3)=w_4\\ L(w_4)=4w_1+3w_2+2w_3+w_4\end{array}$

Calculate coordinate vector of $Lx=Wz$ .

After calculating, we will get

  $Lx=\left[\begin{array}{l}\text{24}\\ \text{25}\\ \text{22}\\ \text{14}\end{array}\right]$

Program:

clc
clear
close all
W = triu(ones(4));
x = (1:4)';
A = [0 0 5 1;
     1 0 4 2;
     0 1 3 3;
     0 0 2 4]; 
y = W^-1*x; 
z = A*y; 
Lx = W*z;
fprintf('The matrix A is: \n')
disp(A)
fprintf('The relation W^-1*x is: \n')
disp(y)
fprintf('The coordinate vector z is: \n')
disp(z)
fprintf('The coordinate vector of Lx is: \n')
disp(Lx)

Quarry:

• First, we have defined the given matrix W.
• Then define the matrix x.
• Calculate coordinate vector of Lx.