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

University of Sargodha

Subject: Software Engineering

BS 3rd Semester/Term Exam 2021

217°3

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 each on your answer sheet.

i.

Define Null space.

(16°2)

ii.

If A is symmetric then show that (A-1) = (AT)

State Calay's Hamilton theorem.

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

coordinate vector relative to bases 5 is (-1,3,2).

xi.

xii.

xili.

XIV.

XV.

xvi.

= e2x are linearly independent

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

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

Define characteristic equation.

Show that matrix A =1

3 -4]

2] is zero of g(x) = x2 + 3x -

10.

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

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

IrA = la 6]. &42 = Lo i finda &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 0

2

7 0

0.2.

6

(a) Compute the determinant of

0 6 3

7 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), uz = (0,1,1) and

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

orthonormal basis.

Q.4.

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(a) Solve the gystem by Gauss elimination method

3x1 + x2-x3 = -4

x_+ X2 - 2x3 = -4

-*+2x-071

(b) Determine whether the vectors in R* ate linear independent or linear dependent

(1,3, - 1, -4), (3,8, -5,7), (2,9,4.23)

0.5.

(a) Consider the set V = Re with landed adition and seir multiplication def goin ,020

for any veR, rER, where F = R. Check whether the set over F form a Vestor space or not?

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

Q.6.

(a) Find the inverse of matrix A

=

1

1

2

(b)

0

Show that the matrix

0

0

0

0

-

1

2

1

cannot be diagonalized.

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