University of Sargodha ADP/BS 2ad Semester (F-24) Examination 2025 Subject: CS/SE/IT Paper: Linear Algebra (MATH-5102)
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University of Sargodha
ADP/BS 2ad Semester (F-24) Examination 2025
Subject: CS/SE/IT
Paper: Linear Algebra (MATH-5102)
Time Allowed: 02:30 Hours
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 each on your answer sheet.
2
i. Define characteristic equation. P
2
iii. For what value of K the vectors (1, -2, K) in R° be a linear combination of vectors (3,0, -2)
and (2, -1, -5). p —
ustani Show thar 2, - 1,%, = et and = gi ane finenty independent by using the Wronakian. P
If A is invertible matrix and n is nonnegative integer, then show that (4") - = (4-*)". P
Show that matrix P is orthogonal if and only if PT is orthogonal. P
7 vit Show that matrix A = [3 3 is zero of 9(2) = 22+ 3x - 10
Ir Band Care both inverses of the matrix 4, then prove that B = c.P ustachi.com
Find a & b, if A = la
ix.
2-x
xi.
xii.
Normalize the vector v = (1,2,4,5). P
Consider the vector 1 = 01, -5,3 and find |ze|loo 1zell 112|12. P
Define Null space. P
Subjective Part
(3*12)
0
0
3
P
Q.2.
(a) Compute the determinant of
2
0.3.
Q.6.
7
3
1
-5.
(b) Show that the set (1,i) in C is linearly independent over R but linearly dependent over C.
(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) Apply the Gram Schmidt process to transform the basis vectors uy = (1,1,1), U2 = (0,1,1) and
Uz = (0,0,1) into an orthogonal basis and then normalize the orthogonal basis vectors to obtain an
orthonormal basis.
(a) Solve the system by Gauss elimination method
P
3x1 + x2 - x3 = -4
x+ x2 - 2x3= -4
-x, + 2x2 - xg =1..
(b) Determine whether the vectors in R* are linear independent or linear dependent
(1,3, - 1, -4) (3,8, -5,7), (2,9,4,23). P
(a) Consider the set V = R" with standard addition and scalar multiplication defined as rv = 0, for any
ER*,rE R, where F= R. Check whether the set Kover F forms a vector space or not? P
(b) Find Eigen values and bases for Eigen spaces of A = 15 - 2] P
(a) IF A = 14
2] then diagonalize that matrix.
(b) Prove that the set (x + iy|x, y are real numbers) forms a Vector Space.
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-- LK-9297/19-11-25
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