NATIONAL ENGINEERING COLLEGE, K.R.NAGAR, KOVILPATTI – 628 503.

BE –Electrical and Electronics Engineering

QUESTION BANK

Course Code / Title :23SH02E- LINEAR STRUCTURES AND TRANSFORMATION

CO 1: Solve the linear system of equations

General system of linear equations – Matrices– Echelon form of matrix- Solving linear systems-
Consistency of a system of linear equations –LU factorization- Applications of system of linear
equations - generating codes with matrices.

CO 1 - PART – A (Two Mark Questions)

Q. No. Text of the Questions Level

1. Find the rank of A=( 12 3 1 4 2 26 5 ) CDL 1

2. Define consistent of system of linear equation CDL 1

3. Find the rank of A=( 2− 21 1 4 − 1 4 6 −3 ) CDL 1

4. Define inconsistent of system of linear equation CDL 1

5. Define system of linear equation CDL 1

6. State unique solution rule for homogeneous equation with example CDL 1

7. State unique solution rule for non homogeneous equation CDL 1

8. State no solution rule fornon homogeneous equation CDL 1

9. State no solution rule for homogeneous equation with example CDL 1

10. State non homogeneous with example CDL 1

11. State homogeneous with example CDL 1

12. Find the solution of the matrix A=( 12 3 0 12 0 0 0 ) CDL 1

13. Find the solution of the system x +2 y − 3 z=− 4 CDL 1

2 x+ y − 3 z=4

−3 y +3 z=12

14. Prove that the system is inconsistent CDL 1

x − 3 y=−7

2 x − 6 y=7
 15. Find the rank of the matrix ( 1 0 0 1 ) CDL 1

CO 1 - PART – B

Q.No Text of the Questions Level Marks

1. Check the consistency of a system of linear equation and discuss CDL 1

the nature of the solution
x 1+ 2 x 2 + x 3=2

3 x 1+ x 2 −2 x3 =1
8
4 x1 −3 x 2 − x3 =3

2 x1 + 4 x 2+ 2 x 3=4

2. Check whether the following system of equations is a consistent CDL 1

system of equations. Is the solution unique or does it have infinite
solutions
x 1+ 2 x 2 − 3 x3 − 4 x 4=6
8
c +3 x 2+ x 3 −2 x 4=4

2 x1 +5 x 2 − 2 x 3 −5 x 4=10

3. Check whether the following system of equations CDL 1

3 x − 2 y +3 z=8

x +3 y+ 6 z=−3 8

2 x+ 6 y+ 12 z=− 6

Is consistent system of equations and hence solve them

4. Check whether the following system of equations CDL 1
x+y + z = 6
3 x −2 y+4z = 9 8
x − y −z = 0
Is a consistent system of equations and hence solve them.
5. Solve the system of equations x1 + x2 + x3 = 1, 3x1 + x2 – 3x3 = 5 CDL 1 16
and x1 – 2x2 – 5x3 = 10 by LU decomposition method.
6. Solve the following equations by LU decomposition method. CDL 1 16
6x1 + 18x2 + 3x3 = 3, 2x1 + 12x2 + x3 = 19, 4x1 + 15x2 + 3x3 = 0
7. Solve the below given system of equations by LU CDL 1
decomposition. 16
x + y + z = 1, 4x + 3y – z = 6, 3x + 5y + 3z = 4

8. Find the solution of the system of equations by LU CDL 1 16

decomposition.
x + 2y + 3z = 9, 4x + 5y + 6z = 24, 3x + y – 2z = 4
 9. Find the rank of the matrix A=( 23 7 3 −2 4 1− 3− 1 ) by reducing it CDL 1 8
to Echelon form
10. Show that the equations x+y+z = 4, 2x+5y-2z =3, x+7y-7z =5 are CDL 1 8
not consistent.
11. Discuss for what values of Ȝ, ȝ the simultaneous equations x+y+z CDL 1
= 6, x+2y+3z =10, x+2y+Ȝz = ȝ have (i). No solution (ii). A 16
unique solution (iii). An infinite number of solutions.
12. Find the values of a and b for which the equations x+y+z=3; CDL 1
x+2y+2z=6;x+ay+3z=b have (i) No solution (ii) A unique solution 16
(iii) Infinite no of solutions.
13. Solve the following system completely CDL 1
x + y + z=1

x +2 y+ 4 z=α 16

2
x +4 y+10 z=α

14. Show that the only real number µ for which the system x+2y+3z = CDL 1
µ x, 3x+y+2z= µ y, 2x+3y+z = µ z, has non-zero solution is 6 and 16
solve them.
15. CDL 1

CO2: Analyze concepts of vector spaces

Vector spaces – Subspaces – Linear combinations – linear span - Linear independence and linear
dependence – Bases and dimensions

CO 2 - PART – A (Two Mark Questions)

Q.No Text of the Questions Level

1. Show that the set Z of all integers with ordinary addition of integers and CDL 1
scalar multiplication as multiplication by any rational number does not
constitute a vector space .
2. Let V be a set of pairs (x,y)of real numbers and let F be the field of real CDL 1
numbers .Define
( x , y ) + ( z , w )=( x+ z , 0 ) ; C ( x , y ) =(cx , 0)

Is V with these operations ,a vector space?

3. Define subspace with example. CDL 1

4. Find, whether or not ,the following vectors are linearly dependent (0,1),(1,0) CDL 1

5. Find, whether or not ,the vectors form a basis ( 1 , 2, −1 )∧(1 , −1 , 5) in R3 CDL 1

6. Show that the vectors three vectors ( 1 ,1 , −1 ) , ( 2 ,− 3 ,5 ) ,(−2 ,1 , 4 ) are CDL 1

linearly independent
 7. Is W ={(a , 0 , 0)/a ∈ R } is a subspace of R3 CDL 1

8. Define linealy independent CDL 1

9. Show that the vectors u=( 1, 2 , 3 ) , v=( 0 , 1 ,2 ) , w=( 0 , 0 , 1 ) generate R3 CDL 1

10. Determine the dimension of the vector space generated by the sets of vectors CDL 1
( 1 ,1 , 0 ) , ( 1 ,0 , 1 ) ,(0 , 1 ,1)
11. Find, whether or not ,the vectors are linearly dependent (4,3),(-3,1) CDL 1

12. If V is the vector space of all (2x2) matrices over R,given a basis of V and CDL 1
dimension of V.
13. Determine the dimension of the vector space generated by the sets of vectors CDL 1
( 1 ,1 , 1 ) , (1 , 0 , 1 ) ,(1 ,2 , 1)
14. Find, whether or not ,the vectors form a basis in CDL 1
3
R ( 1 ,2 , 3 ) , (1 , 0 , −1 ) , ( 3 ,− 1, 4 )∧(2 , 1, 1)
15. Write any two examples of vector space CDL 1

16. Define linealy dependent CDL 1

17. State standard vectors in R3 and prove that they span the vector space R3 CDL 1

18. W be the set consisting of those vectors whose third component is zero. CDL 1
prove that W is a subspace of R3

19. Define the span of a vector space CDL 1

20. Determine the dimension of the vector space generated by the sets of
vectors ( 1 , 0 ,0 ) , ( 0 , 0 , 1 ) , ( 0 ,1 , 0 ) ,(1 ,1 , 1)

CO 2 - PART – B

Q.No Text of the Questions Level Marks

1. Prove that R*R is a vector space over R under usual addition and CDL 1 8
scalar multiplication
2. If W be the set of all triples¿ of real numbers that satisfy the CDL 1
equation 2 x1 − x 2 +3 x3 =C what should be the value of the real 8
number C, if W is to be vector space
3. R*R with addition defined by (a ,b)+(c +d )=(ac , bd ) and usual CDL 1 8
scalar multiplication prove that R*R is not a vector space over R.
4. Given the vectors u= (2,3, -4) & V = (1, -2, -3) is R3. CDL 1
Write (4, -6, -1) as a liner combination of U and V. 8

5. For which the value of K is (1,-2,k) is a linear combination of U & CDL 1 8

V
6. In V 3 ( R ) let e 1=( 1 , 0 ,0 ) ,e 2=( 0 , 1 , 0 )∧e3=( 0 ,0 , 1 ) . CDL 1 8
 7. Determine the vector space spanned by the vectors (3,2,1) & (- CDL 1 8
1,0,1).
8. Show that the threevectors (1,1,-1), (2,-3,5) and (-2,1,4) are CDL 1 8
linearindependent.
9. Examine the linear dependence independence of the following CDL 1
vectors
8
u1=¿ 1, -2,3,4),u2= (− 2 , 4 , −1 , 3 ) u 3=(−1 ,2 , 7 , 6)
.

10. Determine whether the set of vectors (4,1,2,0), (1,2, -1,0) (1,3,1,2) CDL 1
8
& (6,1,0,1) is linearly independent.

11. Show that the vectors U= (1,0,-1), V= (1,2,1) and W= (0,-3,2) form CDL 1
a basis for R3. Express each of the standard basis vectors as a linear 16
combination of U, V, W.
12. Find a basis and the dimension of the subspace W of R4 , generated CDL 1 8
by the vectors (1,-2,5,-3), (2,3,1,-4) & (3,8,-3,-5)
13. If W is the spacespanned by the polynomial V 1=t 3 − 2 t 2 +4 t+1, CDL 1
3 3 2 3 2 16
V 2=t +6 t − 5 ,V 3 =2t − 3 t +9 t − 1∧V 4 =2 t +5 t −7 t +5 ,find a
basis and dimension of W.
14. Given the vectors u= (2,3, -4) & V = (1, -2, -3) is R3. CDL 1
Write (3,8, -5) as a liner combination of U and V. 8

15. Form a basis for the vector space R3 ( 2 , 4 , −3 ) , ( 0 , 1 ,1 )∧(0 , 1 ,− 1) CDL 1

8

CO3:Measure the similarity between different datasets using Inner product spaces
Linear transformation - Null spaces and ranges – Rank Nullity Theorem - Matrix representation of
a linear transformations - Inner product space - Norms - Orthonormal Vectors - Gram Schmidt
orthogonalisation process
. CO 3 - PART –A (Two Mark Questions)

Q.N
Text of the Questions Level
o
1. Define linear transformation with example CDL 1
2. Show that the transformation T : R → R2 defined by T ( x )=(2 x , 3 x) is CDL 1
linear.
3. Find the matrix representation of T on R2 defined by CDL 1
T ( x , y )=(2 y , 3 x − y ) relative to the usual basis.
4. Define orthogonal transformation. CDL 1
5. Find the transpose of the matrix p for the bases of CDL 1
e 1=( 1 , 0 ) , e2= ( 0 ,1 ) ∧f 1= (1 , 3 ) , f 2=(2 , 5)} form e i ¿ f i
6. Define rank and nullity of a linear transformation . CDL 1
7. State any two properties of Eigenvalues. CDL 1
8. Find the transpose of the matrix p for the bases of CDL 1
e 1=( 1 , 0 ) , e2= ( 0 ,1 ) ∧f 1= (1 , 3 ) , f 2=(2 , 5)} form f i ¿ e i
9. Define similarity transformation. CDL 1
10 Find the matrix representation of T relative to the usual basis CDL 1
f 1 ( 1 , 3 ) , f 2= ( 2 , 5 )
11. State Eigenvalues and eigenvectors of a square matrix. CDL 1
12. Define the null space of linear transformation CDL 1
13. Show that the transformation T : R 2 → Rdefined by T ( 1 ,1 )=¿ x − y ∨¿ not CDL 1
linear
14. What is meant by diagonalising a linear operator ? CDL 1
15. Find the matrix representation of T on R2 defined by CDL 1
T ( x , y )=(2 y , 3 x − y ) relative to the usual basis.
16. CDL 1
Define the matrix representation of a linear operator T on a vector space V.
17. When are two matrices said to be similar? CDL 1
18. Find the linear transformation of CDL 1
2
T : R → Rdefined by T ( 1 ,1 )=3∧T ( 0 ,1 )=−2

CO 3 - PART –B

Q.No Text of the Questions Level Marks

1. Show that the transformation T : R 3 → R 2 definedby CDL 1
16
T ( x , y , z )=( z , x + y )is linear
2. Find a linear transformation R3 → R 3whose range space RT is CDL 1
8
generated by (1,2,3) and (4,5,6)
3. Find a linear transformation R4 → R3whose null space N T is generated CDL 1
8
by (1,2,3,4) and (0,1,1,1)
4. If T : R 3 → R 3 is the linear transformation defined by CDL 1
T ( x , y , z )=(2 x + y − z , 3 x −2 y +4 z ), find the matrix of T relative to
16
the bases
e∧f wℎere e ≡ {e 1=( 1 ,1 , 1 ) , e2= (1 , 1 , 0 )∧e 3=( 1 , 0 , 0 )∧f ≡{f 1=( 1 , 3 ) ; f 2=( 1 , 4 ) }
5. If T is the linear operator on R2defined by T ( x 1 , x 2 ) =( − x 2 , x 1 ) , find CDL 1
8
the matrix of T in the basis e ≡ {e 1=( 1 , 2 ) , e2=( 1 , −1 )
6. The matrix A=( 25 − 31 − 4 7 ) determines a linear transformation CDL 1
3 3
T : R → R ,defined by T ( v )= Av , where v is a column vector .show
8
that the matrix representation of T relative to the usual bases of R3
and R2 is itself
7. IfT ( x , y )=( 2 x −3 y , x+ y ) , find ¿verify also that ¿ CDL 1 16
8. Find the eigen values and eigen vectors of the matrix CDL 1
A=( 10 0 0 3 −1 1 −1 3 ) verify that their sum and product are equal to 16
the trace of A and |A| respectively.
9. Verify that the eigen vectors of the following real symmetric matrix CDL 1
8
are orthogonal in pairs A=( 3− 11 −1 5 −1 1− 13 )
10. Diagonalisethe matrix A=( 21 −1 1 1− 2−1 − 21 ) by means of an CDL 1
16
orthogonal transformation .verify your answer
11 Diagonalisethe matrix A=( 2 0 4 0 6 0 4 0 2 ) by means of an orthogonal CDL 1
16
transformation .verify your answer
12. Find the matrix P that diagonalises the matrix ¿ ( 2 21 1 31 1 22 ) by CDL 1 16
 means of similarity transformation.verify your answer.
13. Verify that the eigen values of A2∧ A −1 are respectively the squares CDL 1
and reciprocals of the eigen values of A ,given that 16
A=( 31 4 0 2 6 0 05 )
14. The matrix A=( 25 − 31 − 4 7 ) determines a linear transformation CDL 1
3 3
T : R → R ,defined by T ( v )= Av , where v is a column vector .Find
16
that the matrix representation of T relative to the usual bases of R3
and R2 is itself f ≡ {f 1=( 1 ,3 ) ; f 2=( 2 , 5 ) }
15. Show that the transformation CDL 1
2 2 8
T : R → R definedbyT ( x , y ) =(sin sin x , y) is not linear

CO4: Illustrate Jordan canonical form on a finite dimensional vector space

Generalized eigenvector- Application : Spring and mass in 2D –Chains- Canonical basis the
minimum polynomial- - Algebraic and Geometric multiplicity of Eigen Values - Similar
matricesModal matrix-Jordan canonical form- similarity and Jordan canonical form-Functions of
matrices
CO 4 - PART –A (Two Mark Questions)

Blooms
Q.No Text of the Questions Taxonomy
Level

1 CDL 1
Define Algebraic multiplicity of Eigen Values

2 Find Algebraic multiplicity of( 4 10 4 ) CDL 1

3. Define Geometric multiplicity of Eigen Values CDL 1

4. Find Geometric multiplicity of( 4 10 4 ) CDL 1

5. Determine algebraic multiplicity of A= ( 2 0 0 0 20 0 0 3 ) CDL 1

6. Find Algebraic multiplicity of( 4 0 0 4 ) CDL 1

7. Define similar matrics with example CDL 1

8. Find the minimal polynomial of A= ( 3 2 4 2 0 2 4 2 3 ) CDL 1

9. Determine geometric multiplicity of A= ( 2 0 0 0 20 0 0 3 ) CDL 1

10. Find the minimal polynomial of A=( 1 0 0 1 ) CDL 1

CO 4 - PART –B
Q. No. Text of the Questions Level Marks

1. Find Algebraic and Geometric multiplicity of A= CDL 1 16

( 3 2 4 2 0 2 4 2 3)
2 CDL 1 16
Find the Jordan canonical form of a matrix A= ( 2 21 0 6 2 0 0 2 )
3 Find Algebraic and Geometric multiplicity of A= CDL 1
16
( 1 3 8 0 4 −5 0 0 2 )

4 Find the similar matrix of A= ( 3 1 0− 11 0 1 12 ) CDL 1 16

5 Find the similar matrix ofA= ( 2 21 0 6 2 0 0 2 ) CDL 1 16

6 Find Algebraic and Geometric multiplicity of A= CDL 1

16
( 3 1− 1− 15 −1 1 −1 3 )

7 Find the Jordan canonical form of a matrix A= CDL 1

16
( 3 1 0− 11 0 1 12 )

8 Find the similar matrix of A=( 3 2 4 2 0 2 4 2 3 ) CDL 1 16

9 Find the Jordan canonical form of A=( −2 1 4 − 5 25 −1 1 3 ) CDL 1

.A. What are the eigenvalues and eigenvectors 16
of A? Is Adiagonalizable?

10 Find the Jordan canonical form of A=( 5 4 2 4 5 22 2 2 ).A. What CDL 1

are the eigenvalues and eigenvectors 16
of A? Is Adiagonalizable?

CO5:Decompose the matrix for computational convenience and analytic simplicity

Eigen-values using QR transformations – Generalized Inverse Eigen vectors – Canonical forms –
Singular value decomposition and applications – Pseudo inverse – Moore – Penrose Inverse - Least
square approximations
CO 5 - PART –A (Two Mark Questions)

Q.
Text of the Questions Level
No.

1 Define Penrose Inverse CDL 1

2 Get the singular value decomposition of A=( 22 −1 1 ) CDL 1

3. Explain singular value decomposition in matrix theory CDL 1

4. Prepare a note on least square solution CDL 1

 Solve the system by least square method 𝑥1 + 𝑥2 = 3 , −2𝑥1 + 3𝑥2 = 1
𝑎𝑛𝑑 2𝑥1 − 𝑥2 = 2
5. CDL 1

6. Get the singular value decomposition of A=( 22 −1 1 ) CDL 1

7. Define pseudo inverse of a matrix A CDL 1

8. Summarize the advantage in matrix factorization methods? CDL 1

9. If A is a nonsingularmatrix , then what is 𝐴+ CDL 1

10. Describe the generalized inverse of( 2 2− 22 2 −2 −2 −2 6 ) by least CDL 1

square method

CO 5 - PART –B

Blooms Marks
Q.
Text of the Questions Taxonomy
No.
Level

1 Find the QR decomposition of A=( 11 −1 1 0 01 1 1 ) CDL 1

8

2 Evaluate the singular value decomposition of ( 5 5 −1 7 ) CDL 1 8

Solve the system of equations in the least square sense, 2𝑥 + 2𝑦 − 2𝑧 =

1,2𝑥 + 2𝑦 − 2𝑧 = 3, −2𝑥 − 2𝑦 + 6𝑧 = 2
3. CDL 1
8

4. Find the QR decomposition of A=( 11 −2 0 0 11 0 0 ) CDL 1 8

5. For each of the sets of data that follows, use the least squares CDL 1
approximation to find the best fits with both (i)a linear function
8
and(ii) a quadratic function. Compute the error E in both cases. {(-2,
4), (-1, 3), (0, 1), (1, -1), (2, -3)}

6. Find the singular value decomposition of A=( 2− 1− 21 4 −2 ) CDL 1 8

7. Given any m × n-matrix A (real or complex), the pseudo-inverse A+ CDL 1

of A is the unique n × m-matrix satisfying the following properties: 8
AA+A = A, A+AA+ = A+ , (AA+) $ = AA+ , (A+A) $ = A+A.

Solve the system of equations in the least square sense, 2𝑥 + 2𝑦 − 2𝑧 =

1,2𝑥 + 2𝑦 − 2𝑧 = 3, −2𝑥 − 2𝑦 + 6𝑧 = 2
8. CDL 1 8

9. Foreachofthesetsofdata CDL 1 8
thatfollows,usetheLeastsquaresapproximationto find the best fits with
 both (i) a linear function and (ii) a quadratic function. Compute the
error E in both cases. {(-3, 9), (-2, 6), (0, 2),(1, 1)}

10. Find the singular value decomposition of.( 1 10 0 1 1 ) CDL 1 8