Linear Algebra (LA) BS/MS Software Engineering/PhD Software Engineering 3 Semester/Term University Of Sargodha (UOS) 2021

University of Sargodha

BS 3rd Semester/Term Exam 2021

21703

Subject: Software Engineering

Time Allowed: 02:30 Hours

Paper: Linear Algebra (MATH-201)

Maximum Marks: 80

Note: Objective part is compulsory. Attempt any three questions from subjective part.

Objective Part

(Compulsory)

Q.1. Write short answers of the following in 2-3 lines cach on your answer sheet.

i.

Define Null space.

(16°2)

i. If A is symmetric then show that (A

1) = (4)

1.

State Calay sHamilton theorem

For what Value of K the vectors (1, -2, K) in R° be a linear combination of vectors (3,0, -2) and

(2 FD,-5).

$ = (u = (12,1), = (2,9,0), w = (334)( bases for R°. Find the vector v in R° Whose

oordinate vector relative to bases S is (- 13,4

VI.

vii.

Vili.

IX.

X.

X1.

xii.

xili.

xiv.

XV.

xvi.

Ise Wronskian to show that f. = 1, fRe

and f3 = e2* are linearly independent.

If A is invertible matrix and n is nodnegative integer, then show that (4") -1 = (A-1)"

Show that matrix P is orthogonal if and only if pT is orthogonal.

Define characteristic equation.

Show that matrix A = [3

24] is zero of g(x) = x? + 3x -

10.

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Consider the vector u = (1, -5,3) and find ||u|lo, ||z||. |z||2-

Show that set of all symmetric matrices is subspace of vector space of all n X n matrices.

-]. & 1° = lo i] find a &b.

If B and Care both inverses of the matrix A, then B= C

Write the basis and dimension of vector space V of all Maxmatrices.

Normalize the vector v = (1,2,4,5).

Subjective Part

(3*16)

1

0

3

2

6

Q.2.

(a) Compute the determinant of

0 6 3 0

L7 3 1-5

(b) Show that the set (1,i) in C is linearly independent over R but linearly dependent over C.

Q.3.

(a) Determine whether the vector v = (3,3, -4) is a linear combination of

(1,2,3), y = (2,3,7), z = (3,5,6)

(b) Apply the Gram Schmidt process to transform the basis vectors u, = (1,1,1), u2 = (0,1,1) and

Uз = (0,0,1) into an orthogonal basis and then normalize the orthogonal basis vectors to obtain an

orthonormal basis.

Q.4.

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(a) Solve the system by Gauss elimination method

3x1 + x2- x3 = -4

x, + x2 - 2x3-21.

-4

(b) Determine whether the vectors in Rf are linear independent or linear dependent

(1,3, -1, -4), (3,8, -5,7), (2,9.4,23).

Q.5.

Q.6.

for any ve R"

(b) Find Eigen values and bases for Eigen spaces of A = -2

(a) Find the inverse of matrix A =

0

1

2

N

4.

0

(b)

Show that the matrix

1

2 cannot be diagonalized.

0 0 1]

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