Chapter 3, Problem 1E

(Change of Basis) Set
$U=\text{round}\left(20\ast \text{rand}\left(4\right)\right)-10$ ,
$V=\text{round}\left(10\ast \text{rand}\left(4\right)\right)$
and set $\text{b}=\text{ones}\left(4,1\right)$ .
(a) We can use the MATLAB function rank to determine whether the column vectors of a matrix are linearly independent. What should the rank be if the column vectors of U are linearly independent? Compute the rank of U, and verify that its column vectors are linearly independent and hence form a basis for ${ℝ}^{\text{4}}$ . Compute the rank of V, and verify that its column vectors also form a basis for ${\text{R}}^{\text{4}}$ .
(b) Use MATLAB to compute the transition matrix from the standard basis for ${ℝ}^{\text{4}}$ to the ordered basis $E=\left\{{\text{u}}_{\text{1}},{\text{u}}_2,{\text{u}}_3,{\text{u}}_4\right\}$ . [Note that in MATLAB the notation for the fib column vector ${\text{u}}_j$ is $U\left(:,j\right)$ .] Use this transition matrix to compute the coordinate vector c of b with respect to E. Verify that
$\text{b}=c_1{\text{u}}_{\text{1}}+c_2{\text{u}}_2+c_3{\text{u}}_3+c_4{\text{u}}_4=U\text{c}$
(c) Use MATL.AB to compute the transition matrix from the standard basis to the ordered basis
$F=\left\{{\text{v}}_{\text{1}},{\text{v}}_2,{\text{v}}_3,{\text{v}}_4\right\}$ , and use this transition matrix to find the coordinate vector d of b with respect to F. Verify that
$\text{b}=d_1{\text{v}}_{\text{1}}+d_2{\text{v}}_2+d_3{\text{v}}_3+d_4{\text{v}}_4=V\text{d}$
(d) Use MATLAB to compute the transition matrix S from E to F and the transition matrix T from F to E. How are S and T related? Verify that $S\text{c}=\text{d}$ and $T\text{d}=\text{c}$ .

To determine

Compute the rank of the given matrix.

Answer to Problem 1E

The rank of the matrix is

  $\begin{array}{l}U(rank)=4\\ V(rank)=4\end{array}$

Explanation of Solution

Given information:The matrixes are given

  $\begin{array}{l}U=round(rand(4)\times 20)-10\\ V=round(10\times rand(4))\end{array}$

Given,

  $\begin{array}{l}U=round(rand(4)\times 20)-10\\ V=round(10\times rand(4))\end{array}$

Using row reduce echelon form to compute the rank of the given matrix

The rank of the matrix is

  $\begin{array}{l}U(rank)=4\\ V(rank)=4\end{array}$

Program:

clc
clear
close all
U=round(20*rand(4))-10;
V=round(10*rand(4));
b=ones(4,1);
ru=rank(U);
Urow=rref(U)
rv=rank(V);
Vrow=rref(V)

Query:

• First, we have defined the given matrix.
• Then use the function “rank” to compute the rank of the matrix.
• Get the rank of the given matrix.

To determine

Compute the transition matrix of basis of order 4 for the given matrix U.

Answer to Problem 1E

In the comparison we get the answer, which is close to zero.

  $\text{b=}\left[\begin{array}{l}\text{-0}\text{.2220}\times {10}^{-15}\\ \text{-0}\text{.2220}\times {10}^{-15}\\ \text{ 0}\text{.1110}\times {10}^{-15}\\ \text{-0}\text{.2220}\times {10}^{-15}\end{array}\right]$

Explanation of Solution

Given information:The matrixes are given

  $U=round(rand(4)\times 20)-10$

Given,

  $U=round(rand(4)\times 20)-10$

Using row reduce echelon form to compute the rank of the given matrix

The transition matrix of the given matrix is

  $U=U^{-1}$

Coordinate of b with respect to ordered basis E are given

  $b=ones(4,1)$

Calculate the coordinate vector

  $c=U^{-1}b$

To verify use

  $b=Uc$

Then we get

  $\text{b=}\left[\begin{array}{l}\text{-0}\text{.2220}\times {10}^{-15}\\ \text{-0}\text{.2220}\times {10}^{-15}\\ \text{ 0}\text{.1110}\times {10}^{-15}\\ \text{-0}\text{.2220}\times {10}^{-15}\end{array}\right]$

The above vector is close to zero.

Program:

clc
clear
close all
U=round(20*rand(4))-10;
V=round(10*rand(4));
b=ones(4,1);
ru=rank(U);
Urow=rref(U);
rv=rank(V);
Vrow=rref(V);
d=inv(U);
c=d*b;
b-U*c

Query:

• First, we have defined the given matrix.
• Then use the function “rank” to compute the rank of the matrix.
• Calculate the transition matrix.
• Then use the condition to calculate the coordinate vector.
• Then verify the given relation.

To determine

Compute the transition matrix of basis of order 4 for the given matrix V.

Answer to Problem 1E

In the comparison we get the answer, which is close to zero.

  $\text{b=}\left[\begin{array}{l}\text{-0}\text{.2220}\times {10}^{-15}\\ \text{-0}\text{.4441}\times {10}^{-15}\\ \text{-0}\text{.4441}\times {10}^{-15}\\ \text{-0}\text{.6661}\times {10}^{-15}\end{array}\right]$

Explanation of Solution

Given information:The matrixes are given

  $V=round(10\times rand(4))$

Given,

  $V=round(10\times rand(4))$

Using row reduce echelon form to compute the rank of the given matrix

The transition matrix of the given matrix is

  $V=V^{-1}$

Coordinate of b with respect to ordered basis E are given

  $b=ones(4,1)$

Calculate the coordinate vector

  $d=V^{-1}b$

To verify use

  $b=Vd$

Then we get

  $\text{b=}\left[\begin{array}{l}\text{-0}\text{.2220}\times {10}^{-15}\\ \text{-0}\text{.4441}\times {10}^{-15}\\ \text{-0}\text{.4441}\times {10}^{-15}\\ \text{-0}\text{.6661}\times {10}^{-15}\end{array}\right]$

The above vector is close to zero.

Program:

clc
clear
close all
U=round(20*rand(4))-10;
V=round(10*rand(4));
b=ones(4,1);
ru=rank(U);
Urow=rref(U);
rv=rank(V);
Vrow=rref(V);
d=inv(V);
d=d*b;
b-V*d

Query:

• First, we have defined the given matrix.
• Then use the function “rank” to compute the rank of the matrix.
• Calculate the transition matrix.
• Then use the condition to calculate the coordinate vector.
• Then verify the given relation.

To determine

Show the given relationship between given matrix.

Answer to Problem 1E

The solution is $\begin{array}{l}Sc=d\\ Td=c\end{array}$

Explanation of Solution

Given information:The matrixes are given

  $\begin{array}{l}U=round(rand(4)\times 20)-10\\ V=round(10\times rand(4))\\ b=ones(4,1)\end{array}$

Given,

  $\begin{array}{l}U=round(rand(4)\times 20)-10\\ V=round(10\times rand(4))\\ b=ones(4,1)\end{array}$

Using row reduce echelon form to compute the rank of the given matrix

The transition matrix for E to F is

  $S=V^{-1}U$

And, the transition matrix for F to E is

  $T=U^{-1}V$

Then show

  $S=T^{-1}$

And, show the given condition

  $\begin{array}{l}Sc=d\\ Td=c\end{array}$

Program:

clc
clear
close all
U=round(20*rand(4))-10;
V=round(10*rand(4));
b=ones(4,1);
ru=rank(U);
Urow=rref(U);
rv=rank(V);
Vrow=rref(V);
c=inv(U)*b;
d=inv(V)*b;
S=inv(V)*U;
T=inv(U)*V;
S-inv(T)
d-S*c
c-T*d

Query:

• First, we have defined the given matrix.
• Then use the function “rank” to compute the rank of the matrix.
• Calculate the transition matrix.
• Then use the condition to calculate the coordinate vector.
• Then verify the given relation.