Group assignments given a linear transformation as below, find the f dimension and one basic of im( )

Question 5: Generate a random matrix A which has the size of 4 ×4and is diagonalizable.. A square matrix is said to be diagonalizable if it is similar to a diagonal matrix.. problem: In

VIETNAM NATIONAL UNIVERSITY HO CHI MINHCITY UNIVERSITY OF TECHNOLOGY FACULTY OF

APPLIED SCIENCE -*** -

Lecturer: Phan Thi Huong

Submission date: Saturday, May 14 th , 2022

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● Students work on your assigned groups

● Detailed explanations must be provided to get full scores

● To solve the following questions, please let the constant a be your group ID

● All given exercises must be done by both methods: manual solving

and

using Matlab or Python

TABLE OF CONTENT Project details for each question:

I Theories summary.

II Solution details.

III Coding details.

Question 1: Given a linear transformation f as below, find the dimension

and one basic of Im(f):

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e1 = input('e1 = [a b c]=');

e2 = input('e2 = [a b c]=');

e3 = input('e3 = [a b c]=');

A = [e1', e2', e3'];

%Find the rank of A

- If we are given vector images of a basis E in �㔼ÿ, that

is we know f(E), then where�㔼

ÿ×Āmatrix.

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where f(E), E are two matrices formed by

vector of basis, vector of basis in

Ā = {ÿ1, ÿ2, ÿ., ÿĀ}

- Let f : �㔼Ā

basis for �㔼 Then

is called the matrix representation of f with respect to E

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2.2 Explanation:

Line 1, 2, 3, 4: Create the matrix E, F, x, AEF respectively

Line 5: Find the matrix A through the formula A = F

Line 6: Find f(1;2;9) by using the formula [f(x)]T = A

Question 3: Let F = {(2; -1; 3); (1; 1; 2); (3; 0; 1); (-1; -4; a)} be a subspace of R3

with the inner product < x; y >= 3x1y1 - x1y2 - x2y1 + 4x2y2 + 4x3y3.

a)Find a basis and the dimension of F ⊥

b)Find the vector projection of w = (3; -2; 1) onto F

I Theory:

Definition 1:

Two vectors x , yV in an inner product space V is called orthogonal

<x, y>=0 We denote it by x ⊥ y (x perp y)

Vector x is orthogonal to the set M V if x is orthogonal to every vector in

M We denote it by x ⊥ M

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Theorem 1: Vector x is orthogonal to a subspace F if and only if x is

orthogonal with a basic of F

Definition 2:

A set of two or more vectors in a real inner product space {x1 , x2, …, xn} is

called orthogonal all pairs of distinct vectors in the set are orthogonal

An orthogonal set in which each vector has norm 1 is said to be orthogonal

Scheme of finding F⊥ of a subspace F:

Find a basis of F Assume that basic of F contains vectors {e1 , e2, …, en}

Find the dim and a basic of this null space

Scheme of finding y = projFx

- Find a basis of F, let it be S = {e1 , e2, …, em}

elseif (r == 2)

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E = [dot(e1,e1) dot(e1,e2) dot(e1,e3); dot(e2,e1) dot(e2,e2) dot(e2,e3);

dot(e3,e1) dot(e3,e2) dot(e3,e3)]

D = [dot(x,e1); dot(x,e2) ; dot(x,e3)];

X = mldivide(sym(E), sym(D)) ;

fprintf('The projection of w onto F: ')

(X(1,1)*e1 + X(2,1)*e2 + X(3,1)*e3)

F =

2-1

Fo =

5-4

7dimFo =

0

E =

1479

The projection of w onto F:

ans=

[ 3, -2, 1]

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Question 4: Determine the currents I1, I2, I3 and for the given electrical network.

1 Solution:

Eauation (1), (2) & (3) in

clc, clear, close all;

disp('Solve the systems AX = B using gaussian

elimination'); n = input('Input number of equations n = ');

k = input('Input number of variables k = ');

if result(i,i) == 0

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disp('Infinity Solution');

disp([num2str(k - rresult) ' free variable']);

Question 5: Generate a random matrix A which has the size of 4 ×4

and is diagonalizable

Then, return its eigenvalues, the corresponding eigenvectors and compute A100.

1 What is a diagonalizable matrix?

A square matrix is said to be diagonalizable if it is similar to a diagonal matrix

That is, A is diagonalizable if there is an invertible matrix P and a diagonal matrix

D such that A=PDP -1

2 What are eigenvalues and the corresponding eigenvectors of a matrix?

Many problems present themselves in terms of=an eigenvalue problem:

In this equation A is an n-by-n matrix, v is a non-zero n-by-1 vector and λ is a

scalar (which may be either real or complex) Any value of λ for which this

equation has a solution is known as an eigenvalue of the matrix A It is

sometimes also called the characteristic value

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The vector, v, which corresponds to this value is called an eigenvector The

eigenvalue problem can be rewritten as:

This equation is called the characteristic equation of A, and is an nth order

polynomial in λ with n roots These roots are called the eigenvalues of A

We will only deal with the case of n distinct roots, though they may be repeated For each eigenvalue there will be an eigenvector for which the eigenvalue equation is true This is most easily demonstrated by example:

If �㔼 = [

Then the

characteristic equation is

And the two eigenvalues are:

All that's left is to find the two

associated with the eigenvalue, λ1=-1, first

so clearly from the

Note that if we took the

In either case we

which the two elements have equal magnitude and opposite sign

where k

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could have used any two quantities of equal magnitude and opposite sign.

Going through the same procedure for the second eigenvalue:

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2 = 222 ].[+1

Again, the choice of +1 and -2 for�the eigenvector� was arbitrary; only their ratio is

important This is demonstrated in the MatLab code below

Diagonal matrices are relatively easy to compute with, and similar matrices sharemany properties, so diagonalizable matrices are well-suited for computation

In particular, many applications involve computing large powers of a matrix,

Choose the ma trix �㔼

The unit matrix is � 㔼

Then we have

So that

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Solve this equation we have:

,which are the eige nva lue s of the matrix �㔼 2

5 Coding details

a Full MATLAB code:

%Clear all text from the Command Window

clear all;

clc;

%Input the matrix 4x4 that we need to use in this problem

A=input('Enter the matrix A: ')

%Check the matrix if it is already diagonal?

%Show the output of the problem: V is the matrix concluding all

%eigenvectors and D is a diagonal matrix with eigenvalues located

on the %main diagonal

%Ask the user to input another matrix

disp("The matrix is not diagonalizable Let's try another

FIGURE 1

Howeve r, if we input a ma trix tha t is not diagona lizable �㔼

The result will be:

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