matlab bài tập lớn đại số tuyến tính

Currently, science is growing, with this development strength, the application of science and scientific patents in schools is very feasible and significant. Since the first year, Ho Chi Minh City University of Technology lecturers have helped technical students get acquainted with programming applications such as MATLAB. MATLAB is a rising and programming environment that allows calculating numbers with matrices, graphing functions or charting information, implementing algorithms, creating user interfaces, and linking to computer programs written in many other programming languages. MATLAB enables simulation of calculations with the Toolbox library, experimenting with many models in practice and techniques. With more than 40 years of establishment and development, MATLAB is an effective calculation tool to solve technical problems today with a relatively simple and universal design. Therefore, for the problems in Algebra, especially matrix problems, we can use MATLABs computing applications to solve most simply and understandably, helping us get acquainted and add more skills to use programs application for students.

HCMC University of Technology MATLAB Exercises

Department of Applied Mathematics Subject: Linear Algebra

1 Introduction

Currently, science is growing, with this development strength, the

application of science and scientific patents in schools is very feasible and significant Since the first year, Ho Chi Minh City University of Technology lecturers have helped technical students get acquainted with programming applications such as MATLAB MATLAB is a rising and programming

environment that allows calculating numbers with matrices, graphing

functions or charting information, implementing algorithms, creating user interfaces, and linking to computer programs written in many other

programming languages MATLAB enables simulation of calculations with theToolbox library, experimenting with many models in practice and techniques.With more than 40 years of establishment and development, MATLAB is an effective calculation tool to solve technical problems today with a relatively simple and universal design Therefore, for the problems in Algebra,

especially matrix problems, we can use MATLAB's computing applications to solve most simply and understandably, helping us get acquainted and add more skills to use programs application for students

3 Solutions

Question 1: Find the argument, module of z=1+i√3

1+i z= 1+i√3

Question 2: Find the argument, module of z = (1+i√ 3) (1-i)

z2=z−´z ↔ (x + yi)2=x+ yi−(x− yi) ↔ x2

+2 xyi− y2=2 yi

↔ {x2−y2=0

2 xy=2 y ↔ {x=±1 x=1 , {x=0 y=0

The solution to the complex equation is z=0, z=1-i and z=1+i

Question 6:A=(2 −1 42 1 3 −1)5 ; B=(−1 3 0 −1)1 2 0 −1 Find C = A T B, trace of C, rank of C, det of C.

A=(2−1 4 521 3−1); B=(−1 3 0−11 26−1 )

−10)A=(−10 2 −4

12

Det (AB)= [-9(2+m)-208m+27(-13+6m)]-[-4(-13+6m)]-117(2+m)+108m]≠0m≠−172

8

−78

−18

−118

98

1413

8

−78

−18

−11

8

98

−1

8 ) → P A=det ( A ) A−1=8( −54

34

1413

8

−78

−18

−118

98

−1

8 )=(−1013 −7 −16 2

−11 9 −1)

Question 16: Reduce the matrix (1 1 2 12 3 4 5

3 4 6 9) to the row echelon form

6)−r2+r3→r3

→ (1 1 20 1 0

0 0 0

13

3)

Question 17: Solve the equation (3 −15 −2)X=(5 67 8)

(3 −15 −2) X=(5 67 8)

1641

Question 19: Find the NUMBER of solutions of this system ¿

0 0 4 5

0 0 0 0 |−74

5

0 ) The system corresponding to this matrix is:

⇒ The system has no solution

Question 22: Find m such that the following system has a unique solution

Question 25: Find the dimension and one basis of the span

Question 26: Given V =¿(1 ; 2 ; 1; 1) , (2;−1 ; 1; 3 ),(5 ; 5 ; 4 ; m)>¿ Find m such that

dim(V) is max find a basis of V.

V=(12 −1 12 1

5 5 4

13

m−5)−r2+r3→ r3

→ (10 −5 −12 1

11

→ The given requirement is equivalent to such that det(A) ≠ 0 ↔

Question 33: In R3, given 2 bases E={(1;0;1),(1;1;1),(1;1;0)} and

E’={(1;1;2),(1;2;1),(1;1;1)} Find the change of basis matrix from E to E’

Question 34: Find m such that x = (1; 0; m) is a linear combination of M =

→ m does not exist such that G ⊂F

Question 36: In R4, given a subspace

V ={(x1; x2; x3; x4)∈ R4∨x1−x2+x3=0∧ x2+x3+x4=0

Find the dimension, one basis of V

Question 40: In R3 with the dot product, given u=(1 ;1;2), v=(2 ;1 ;−1) Find

ⅆ(u , v ) and find a vector ω orthogonal to u, v

In R3, given a dot product: (u , v )=u1v1+u2v2+u3v3

u−v=(−1 ;0 ;3) The distance between u and v is d (u , v)=‖u−v‖=√ ⟨u−v ,u−v⟩ =

√(−1) (−1)+0+3.3 = √10

Cross product: |i ⃗j⃗ ⃗k

1 1 2

2 1 −1|=−3 ⃗i+5 ⃗j−⃗k A vector ω orthogonal to u , v is (-3;5;-1).

Question 41: In R3, given an inner product

( x , y )=2 x1y1−3 x1y2+x2y2−x2y3−x3y2+4 x3 y3

Find the distance between 2 vectors u=(1 ;2;1) and v=(−1;1 ;2)

u = (1,2,1) v = (-1,1,2) A=(−32 −35 −10

0 −1 4 )

d(u,v) = √(u−v ) A (u−v )T=√7

Question 42: In R3, given an inner product

Question 43: In R3, given an inner product

The basis of Imf is {(1;0;1),(0;1;-1)} and dim ( Imf )=2.

Question 46: Given f: R3→ R2 satisfying f (1 ;1; 0)=(2;−1), f (1 ;1;1 )=(1 ;2),

Therefore, the eigenvectors corresponding to λ1 = -4 are β (−65 , 1), β ≠0.

⇒u=(−56 ) is the eigenvector of A

Question 50: Given A=(3 64 5), λ1−1, λ2=3 Which number is an eigenvalue of

Question 51: Given A=(3 1 12 4 2

1 1 3) Find all eigenvalues and the associated eigenvectors of A

Given A=(3 1 12 4 2

1 1 3) Find all eigenvalues∧theassociated eigenvectorsof A

* Find the eigenvalues

( A−I ) x=0 ↔ det ( A−I )=0 ↔| (3 1 12 4 2

So the eigenvalues is 2 and 6 and the eigenvectors of A is (−1 −1 11 0 2

0 1 1)

Question 52: Given A=(−10 −8−8 67

1 −14 m) Find m such that A has an eigenvalue

λ=2 Find all eigenvalues and the associated A with m found

Question 54: In R3, given M = {(1; 2; −1), (3; 2; −1), (0; 2; −1)} Find m

such that (3; 8; m) is a linear combination of M

Question 56: In R4, given U =¿(1,2,1,1); (2,1,0 ,−2)>¿ and

V =¿(1,5,3,5 ); (3,0,−1 , m)>¿ Find m such that U ≡V

Because U≡W the m=-5

Question 57: In R4, V is the nullspace { x1+x2−x3=0

2 x1+2 X2+x3+x4=0

x1+x2+2 x3+m x4=0Find m such that dim ⁡(V ) is max Find the dimension and a basis of V with m inthe question a

dim V=2 and basis= [1, 1, -1,0], [0,0,3,1]

Question 58: In R4, given U =⟨(1,2,1,0 ); (2,−1,1,1)⟩V =⟨(1,1 ,−2,1) ;(2,0,4 , m)⟩ Find msuch that dim(U+V) is min Find the dimension, a basis of U ∩V

→ x=α1 (1,2,1,0)+α2 (2 ,−1,1,1)=α2.(35,−

19

5 ,−

25,1)

→ dimensionof U ∩V is1∧a basic is{ (35,−

19

5 ,−

25,1) }

Question 59: In R4 , given 2 nullspaces

U :[ 1 1 2 0

−1 1 −1 2|00], V : [ 1 2 2 2

−1 0 −1 m|00].Find m such that dim(U ∩ V ) is max Find the dimension, a basis of U ∩ V

Then the basic of V⊥ is V⊥ = {(1 ,−1,1,0),(0 ,−2,0,1)}

Question 62: In R4, given a span V =¿(2;−1; 1 ; 0) , (−2 ; 1; 0 ; 1)>¿ and a vector

x=(1;1 ;0 ;1) Find Pr v( x)

Prv(x)= V*(VT *V)-1*VT *x

Question 64: In R3 with the standard basis, given F = ¿(1 ; 1; 2) , (2; 1 ;−1)>¿ and

a vector x = (1; 2; 3) Find the projection of x onto F

35 )

Question 65: In R3, given an inner product ( x , y )=x1y1+2 x2y2+3 x3y3−x1y3−x3y1

Find the angle and the distance between u=(1 ;1;2) and v=(2 ;1 ;−1)

The given two vectors are u=(1 ;1;2) and v=(2 ;1 ;−1) in R3

Question 66: In R3, given an inner product

( x , y )=x1y1+2 x2 y2+5 x3y3−2 x1 y3−2 x3y1 Find the orthogonal complement of

Question 70: Given a linear transformation f: R3→ R2, and the matrix of f in

E={(1 ;1;1) , (1;0 ;1) , (1;1 ;0) }, F={(1 ;1) ,(2 ;1) } is A E , F=(2 1 −30 3 4 ) Find the matrix of

f in the standard bases

Question 71: Given a linear transformation f: R3

→ R3 with the matrix in the basis E={(1 ;2;1) , (1;1 ;2) ,(1 ;1;1) } is

A E=(1 0 12 1 4

1 1 3) Find the matrix of f in the standard bases

We call the matrix basis of f is e

Question 73: Given a linear transformation f : R3

→ R3 with the matrix in the basis E={(1 ;1;2) , (1;1 ;1) ,(1 ;2;1) } is

[f (1;2 ;1)] E=(1 1 −12 3 3

1 2 4 ) (00

1)=(−13

4 ) → f (1 ;1;2 )=−(1 ;1;2)+3 (1 ;1;1)+4 (1;2 ;1)=(6 ;10 ;5) Imf =¿f (1 ;1 ;2), f (1 ;1 ;1 ), f (1 ;2;1)≥¿(4 ; 5 ; 5) , (6 ; 8 ; 7) ,(6 ; 10 ; 5)>¿

A basis of Imf is {(4;5;5), (0;-2;2)} and the dimension of Imf is 2.

Question 74: Given a linear transformation f : R3 → R3 with the matrix in thebasis E = {(1; 0; 1),(1; 1; 1),(1; 1; 0)} is

Question 2: Find the argument, module of z = (1+i√ 3) (1-i)

[x,y] = solve( 'x^2 - y^2 - x= 0' , '2*x*y + y = 0' );

disp( 'The solution to the complex equation is ' );

[x,y] = solve( 'x^2 - y^2 = 0' , '2*x*y - 2*y = 0' );

disp( 'The solution to the complex equation is ' );

12

%compute f(A) using the formula

f_A = A^2 -(2*A) -3

Question 16: Reduce the matrix (1 1 2 12 3 4 5

3 4 6 9) to the row echelon form

Question 20: Find the NUMBER of solutions of this system

Question 26: Given V =¿(1 ; 2 ; 1; 1) , (2;−1 ; 1; 3 ),(5 ; 5 ; 4 ; m)>¿ Find m such that

dim(V) is max find a basis of V.

ans =

[ 1, 0, 0]

[ 0, 1, 1]

[ 0, 0, 0]

% Since the rref (row reduced echelon matrix) of the coefficient matrix of

% given vectors has a row with all elements zero So, the rref is not a

% full rank matrix (it doesn't matter what have we chosen 'm') Hence,

% given vectors are linearly dependent for all 'm' Therefore, the given

% set cannaot be a basis of P_2[x].

% There does not exist such 'm' for which the given set is a basis for

% P_2[x].

Question 33: In R3, given 2 bases E={(1;0;1),(1;1;1),(1;1;0)} and

E’={(1;1;2),(1;2;1),(1;1;1)} Find the change of basis matrix from E to E’

>> F=[1 1 -1 -1;1 -1 3 -1]

F =

>> %m does not exist

Question 36: In R4, given a subspace

Question 40: In R3 with the dot product, given u=(1 ;1;2), v=(2 ;1 ;−1) Find

ⅆ(u , v ) and find a vector ω orthogonal to u, v

u=[1 1 2];

v=[2 1 -1];

d=norm(u-v)

w=cross(u,v)

Question 41 & 42: In R3, given an inner product

Question 45: Find the dimension, one basis of Imf:

Question 50: Given A=(3 64 5), λ1−1, λ2=3 Which number is an eigenvalue of

1 −14 m) Find m such that A has an eigenvalue

λ=2 Find all eigenvalues and the associated A with m found

Question 56: In R4, given U =¿(1,2,1,1); (2,1,0 ,−2)>¿ and

V =¿(1,5,3,5 ); (3,0,−1 , m)>¿ Find m such that U ≡V

Question 59: In R4 , given 2 nullspaces

U :[ 1 1 2 0

−1 1 −1 2|00], V : [ 1 2 2 2

−1 0 −1 m|00].Find m such that dim(U ∩ V ) is max Find the dimension, a basis of U ∩ V

Question 64: In R3 with the standard basis, given F = ¿(1 ; 1; 2) , (2; 1 ;−1)>¿ and

a vector x = (1; 2; 3) Find the projection of x onto F

F=[1 1 2;2 1 -1]';

x=[1 2 3]';

PrFx=F*inv(F'*F)*F'*x

Question 65: In R3, given an inner product ( x , y )=x1y1+2 x2y2+3 x3y3−x1y3−x3y1

Find the angle and the distance between u=(1 ;1;2) and v=(2 ;1 ;−1)

Question 66: In R3, given an inner product

( x , y )=x1y1+2 x2 y2+5 x3y3−2 x1 y3−2 x3y1 Find the orthogonal complement of

F=¿(1 ;2 ;3)>¿

Question 70: Given a linear transformation f: R3→ R2, and the matrix of f in

E={(1 ;1;1) , (1;0 ;1) , (1;1 ;0) }, F={(1 ;1) ,(2 ;1) } is A E , F=(2 1 −30 3 4 ) Find the matrix of

f in the standard bases

Question 72: Given a linear transformation f: R3→ R3 with the matrix in the basis E={(1 ;2;1) , (1;1 ;2) ,(1 ;1;1) } is A E=(1 0 12 1 4

1 1 3) Find the matrix of f in

Question 75: Write a program to find: Input: matrix A Count the number

of even entries of A

% ask user to enter rows and columns

m=input( 'Enter the rows: ' );

n=input( 'Enter the columns: ' );

% ask user to enter the elements of the martix

fprintf( 'Total even entries are: %d' ,evenEntries)

Question 76: Write a program to find: Input: matrix A Find the

maximum elements in each row of A

% ask user to enter rows and columns

m=input( 'Enter the rows: ' );

n=input( 'Enter the columns: ' );

% ask user to enter the elements of the martix

% Find the maximum element in each row of A

disp( 'The maximum elements in each row are :' )

max(a,[],2)

Question 77: Write a program to find: Input: matrix A Find the

maximum positive entry in A.

% ask user to enter rows and columns

m=input( 'Enter the rows: ' );

n=input( 'Enter the columns: ' );

% ask user to enter the elements of the martix

% check for all positive elements and find maximum

% consider 0 to be positive as well

% ask user to enter rows and columns

m=input( 'Enter the rows: ' );

n=input( 'Enter the columns: ' );

% ask user to enter the elements of the martix

% check for all odd elements

% and add them to a sum

% print the required answer

fprintf( 'Sum of Odd Elements is %d' ,sumOfOddEntries)

Question 79: Write a program to find: Input: matrix A Find the product

of all odd entries in A.

% ask user to enter rows and columns

m=input( 'Enter the rows: ' );

n=input( 'Enter the columns: ' );

% ask user to enter the elements of the martix

% print the required answer

disp( ' The Product of all elements in a matrix is' );

disp(product);

Question 80: Write a program to find: Input: matrix A Check if A is

square and symmetric

% ask user to enter rows and columns

m=input( 'Enter the rows: ' );

n=input( 'Enter the columns: ' );

% ask user to enter the elements of the martix

disp( 'Matrix A is neither square nor symmetric' );

% ask user to enter rows and columns

m=input( 'Enter the rows: ' )

n=input( 'Enter the columns: ' )

% ask user to enter the elements of the martix

fprintf( 'Orthogonal Matrix exists for only Square Matrix' )

% if matrix product of A and B forms identity matrix it is Orthogonal

elseif A*B==I

fprintf( 'The given matrix is Orthogonal' )

% in all other cases it is not Orthogonal

else

fprintf( 'The given matrix is Not Orthogonal' )

end

-The