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Answered) Assignment NO. 3 week9-week11 Student Full Name:___________________________________ . Student ID:__________________________________________ . CRN...

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Assignment NO. 3 week9-week11

Student Full Name:___________________________________ .

Student ID:__________________________________________ .

CRN No:____________________________________________ .

Branch: _____________________________________________. Linear Algebra

(Math-251) Total Points

True/False ____/6 MCQ ____/6 Short Answer ____/18 Total ____/30 1|7Page Due date: 05 May 2017 Max Marks: 30 Section-I

State whether the following statements are true or false: [6X1=6] 1. Matrix is diagonalizable if it is similar to a diagonal matrix B;

that is, there exists an invertible matrix and = âˆ’1 .

(1)â€¦â€¦â€¦â€¦..

1 0

2. The eigenvalues of the matrix A =[âˆ’2 3

0 4

7. 0

0 ], are 1, 3, and âˆ’7

(2)â€¦â€¦â€¦â€¦.. 3. A square complex matrix is called Unitary if its conjugate

transpose equal to the matrix . (3)â€¦â€¦â€¦â€¦.. 4. The inner product of a nonzero vector with itself (âŒ© u, u âŒª ) is

always a positive real number. (4)â€¦â€¦â€¦â€¦.. 5. If = (1,2, âˆ’3,4) and = (âˆ’2,1,4,3) then âŒ©, âŒª = 0 , where

âŒ© , âŒª denotes the Euclidean inner product.

4

6. The matrix A = [2 âˆ’ 2 (5)â€¦â€¦â€¦â€¦.. 2âˆ’

2

3

âˆ’3 âˆ’ 6 ] is Hermitian.

6 âˆ’ 3

2

(6)â€¦â€¦â€¦â€¦.. 2|7Page Section-II

For Each Question, Choose the Correct Answer from the MultipleChoice:

[6X1=6]

1. Which of the following sets of vectors are orthogonal with

respect to the inner product, defined by âŒ© u, v âŒª = 21 1 +

52 2 on R2:

a. (2,5), (8,-3)

b. (3,4), (2,6)

c. (5,3), (3,-2)

d. (1,5), (3,-2)

2. The values of for which = (, 7, 2), and = (, âˆ’, 5) are

orthogonal in R3 with respect to Inner Product âŒ© u, v âŒª = 1 1 +

2 2 + 3 3 , are

a. 2, -5

b. -2, 5

c. -2,- 5

d. 2, 5

3. If 0 is an eigenvalue of a square matrix A then A is:

a. an Identity matrix.

b. invertible.

c. not invertible.

d. None

4 + 2i 1 + 8

4. If V = [ 2 âˆ’ 2i

1 âˆ’ 3i] , a complex matrix, then âˆ— is

8

6 âˆ’ 7

âˆ’5

4 âˆ’ 2i 1 âˆ’ 8

a. [ 2 + âˆ’2i

1 + 3i]

8

6 âˆ’ 7

âˆ’5

4 âˆ’ 2i 2 + 8

b. [ 1 âˆ’ âˆ’2i

âˆ’6i âˆ’ 7]

8

1 + 3

âˆ’5

4 âˆ’ 2i 2 âˆ’ 8

c. [ 1 + âˆ’2i

âˆ’7 âˆ’ 6i]

8

1 + 3

âˆ’5

4 + 2i 1 + 8

d. [ 2 âˆ’ 2i

1 âˆ’ 3i].

8

6 âˆ’ 7

âˆ’5

3|7Page 5

5. The characteristic equation of the matrix A = [

4

a. 2 âˆ’ 2 âˆ’ 19 = 0

b. 2 + 2 âˆ’ 19 = 0

c. 2 âˆ’ 2 + 19 = 0

d. 2 + 2 + 19 = 0 1 is

âˆ’3 6. Cosine value of the angle between vectors u and v is 1 and

|||| = 3, |||| = 5, then âŒ© u, v âŒª =

a. 10

b. 15

c. âˆš15

d. âˆš10

Answer:

1 2 3 4 5 6 Section-III

Attempt all the questions: [6X3=18] 1. Find the complex dot product âˆ™ , such that = (5, âˆ’2, 2 +

, 4), = (3, 1 + 2, 1,3).

Solution: 4|7Page 2. Find all least square solutions and the error vector of the linear

system:

1 âˆ’ 2 = 2

1 + 2 = 4

21 + 2 = 8

Solution: 3. Show that matrix A = 1 âˆ’2 2 3

2 3

âˆ’1 3

âˆ’2 3

2 3

2 3

1 3 3 [3

Solution: 5|7Page is orthogonal and find A-1 4. Find the eigenvalues and the eigenvectors of the matrix

7 âˆ’4

=[

5 âˆ’2

Solution: 5. Show that matrix + is Hermitian , where =

0

0 3

1

âˆ’ 2

[ âˆ’2 0 ] , = [ 0

âˆ’1 1] .

2âˆ’ 0

1

âˆ’2 1 1 Solution: 6|7Page 6. Verify the property âŒ© u + v, w âŒª = âŒ© u, wâŒª + âŒ© v, w âŒª, If the inner

product on 2 , defined by âŒ© u, v âŒª = 21 1 + 32 2 .

Solution: 7|7Page

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