Linear Algebra/Linear Algebra (LA) BS/MPhil Computer Science/MS Computer Science 4 Semester/Term UOS — University of Sargodha 2024

Define trace of a matrix.

ii.

ii.

iv.

vi.

Define Markove Matrix

vii.

Find inverse of A =

vili.

x.

XI.

Write the basis of Pn (x).

Define similar matrices.

University of Sargodha

Subject: Computer Science

Time Allowed: 02:30 Hours

BS 4th Term Examination 2024

Paper: Linenr Algebra (MATH-202/MATH-3215)

For Regular & Retake Students)

Maximum Marks: 60

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 of cuch on your anywer sheet.

Whether the vectors 4 = (1,2, -3), u½ = (1, -4,3), are orthogonal or not.

Write the bases for the vector space M2x2 of 2 × 2 matrices.

Let V be vector space and u EV then show that (-1)u = -u.

v. It A is invertible matrix, then AT is also invertible and (AT)-1 = (4-1)T.

(2°

12)

32

4

Let V be a vector space over a field K. Show that for any scaler k and 0 € V, KO = O.

Show that set of all matrices with trace zero is subspace of vector space of all 71 × n matrices.

] show that At =

12.

xii.

Subjective Part

(3"12)

Q.2.

(a) Determine whether (1,1,1,1), (1,2,3,2), (2,5,6,4). (2,6,8,5) form basis of R*. If not, find the

dimension of the subspace they span.

(b) Show that matrix A = [3

4 satisty its characteristic equation.

(a) Let W be subspace of R° spanned by the vectors u1 = (1,2, -1,3), u½ = (2,4,1, -2),

U⅓ = (3,6,3, -7), u; = (1,2, -4,11), us = (2,4, -5,14). Find busis and dimension of W.

(b) Find Eigen values and corresponding Eigen vectors of A = [3 1]

Q.4.

(a) Consider the vectors u; = (1,2,1,3,2), ½ = (1,3,3,5,3), Ug = (3,8,7,13,8),

W1 = (1,4,6,9,7), W≥ = (5,13,13,25,19) in R*, let U = span(u), w = span(wl).

Then show that U = W

(b) Solve the following system of Linear equations by using Row Operation

x+y+2：=9

2x+4y-32=1

Зx+бу-5z=0

2

دما

Q.5.

Q.6.

（2） Find f

, if A=

-20

(b) Apply the Gram-Schmidt process to find an orthogonal basis and then an orthonormal basis for

t the subspace U of R* spanned by 11 = (1,1,1,1), uz = (1,2,4,5), ug = (1, -3, -4, -2).

(a) IF A = [3

-1] then diagonalize that matrix.

(b) Let v, = (1,2,1), v, = (2,9,

0) and v, = (3,3,4). Show that the set S = {V, ½,%,} is basis for R3.