University of Sargodha BS 3rd Term Examination 2024 Subject: Software Engineering/IT

University of Sargodha

BS 3rd Term Examination 2024

Subject:

Software Engineering/IT

Paper:

Linear Algebra (MATH-3215/MATH-203/MATH-201)

Time Allowed: 02:30 Hours

Maximum Marks: 60

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

Objective Part

(Compulsory)

us a dirie short answers of the foowing in 2-3 lines cuch on your anster tet.

(2*

12)

i. Find inverse of A = l4

ii. Define Eigen value.

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Whether the vectors u, = (1,2

-3), 42 = (1, -4,3), are orthogonal or not.

iv. Consider vectors in R° u = (2,1,1), = (12, -3) and w = (1, -43) then which vectors are

v. Write the bases for the vector space M2x2 of 2 x 2 matrices.

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

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vil. If A is invertible matrix then AT is also invertible and (AT) -1 = (A-1).

vili. Define Rank and Nullity of homomorphism.

2+y 22+1=l5l

x. Define trace of a matrix.

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

0.

xii. Show that set of all matrices with trace zero is subspace of vector space of all n X n matrices.

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 4 = l3

4J

satisfy its characteristic equation.

0.3.

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

subspace U of R* spanned by u, = (1,1,1,1, u2 = (1,2,4,5), u3 = (1, -3, - 4, -

2)

b) Find Eigen values and coresponding Eigen vectors of A = I!

a) Consider the vectors u, = (1,2,1,3,2), 42 = (1,3,3,5,3), uz = (3,8,7,13,8), W, = (1,4,6,9,7),

W½ = (5,13,13,25,19 ) in R*, let U = span(ul), w = span(w). Then show that U = W

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

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x+y+2z=9

2x+ 4y-32=1

3x+6y - 5==0

a) Let W be subspace of R° spanned by the vectors u, = (1,2, - 1,3), u2 = (2,4,1, - 2),

Ug = (3,6,3, - 7), u, = (1,2, -4,11), ug = (2,4, -5,14). Find basis and dimension of W.

2

b) Find A

, if A=

-2

0

3

0

37

2

-4

2

Q.6.

-1

then diagonalize that matrix.

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b) Let v, = (1,2,L), V, = (2,9,0) and v, = (3,3,4). Show that the set S= (y, %,,v,) is basis for R3.

-- LK-6607/14-05-24

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