Linear Algebra ADP/BS 2 Semester/Term UOS — University of Sargodha 2025

3 - ‘ Q a Yr : 0 d h a University of: » ’ nd ’ pe = ADP/BS 2" Semester (E VAT 5102 Subject: CS/SEAT Paper; BIBES=2 Maximum Marks: 60 Time Allowed: 02:30 Hours =. (J art. 3 Note: Objective part is compulsory. Attempt any three ¢ gestions from ubjectie Objective Part (Compulsory) Q.1. Write short answers of the following in 2-3 1; heet (2*12) : gn 2-3 lines our answer s . L Define characteristic equation. - nes di. If'A is symmetric then show that(4~1)T = Aan. : 3.0, —2) ni. For what value of K the vectors (1, =2 8) in R3 be a linear combination of vectors (3,0, —<, and (2,-1,-5). : iv. Show that f; = 1,f, = e* and fj = ?* are linearly independent by using the Wrons iggy v. If Ais invertible matrix and nis nonnegative integer, then show that (A")"" = (AT) vi. Show that matrix P is orthogonal if and only if PT is orthogonal. vii. Show that matrix A = E | is zero of gx) = x? + 3x — 10. viii. Finda&b, ifA =[> J], aa? = 5 il ix. If B and C are both inverses of the matrix A, then prove that B = C. X. Normalize the vector v = (1,2,4,5). } xi. Consider the vector u = (1, —5,3) and find llelle , [l2¢ll1, lell2- ADP/E Subject: CS/SEAT Maximum Marks: 60 Note: Objective part is compulsory. Attempt any three § sestions from subjective p Objective Part ¢ Compulsory) (2*12) Q.1. Write short answers of the following in 2-3 lines each on your answer sheet. i Define characteristic equation. I. AFA is symmetric then show that(A=1)T = (47). Ni For what value of K the vectors (1,=2 K) in R? be a linear com and (2,-1,-5). iv. Show that f; = 1, f> = e* and f3 = e2* gre linearly independent by usin v. If Ais invertible matrix and nis nonnegative integer, then show fst Cb vii. Show that matrix 4 = E A is zero of g(a) = x? + 3x — 10. viii. Finda&b, ifd =! Elles = 5 El. X. Normalize the vector v = (1,2,4,5). 1} xi. Consider the vector u = (1, —5,3) and find Jello , lll1, Nwell2- xii. Define Null space. bination of vectors (3, 0,—2) g the Wronskian, Yyl= A) Subjective Pa (3*12) 1 Ont 3 ’ 2 7S 6 £4 Compute the determinant of ) 2 © 3 0 6 3 0 7 SES (b) Show that the set {1,7} in Cis linearly independent over IR but linearly dependent over C. ‘ Q.3. (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. i K (b) Apply the Gram Schmidt process to transform the basis vectors uy = (1,1,1),u, = (0,1,1) and us = (0,0,1) into an orthogonal basis and then normalize the orthogonal basis vectors to obtain an orthonormal basis. BN Q4. (a) Solve the system by Gauss elimination method ~L\ 3x +x, — x3 = —4 A L¥ Xy tx — 2x; = —4 i \ f FE |& - 2X; — X9°53 15. C6 Determine whether the vectors in R* ar independent or linear dependent (1,3, -1, —4), (3,8, =5,7), (294,23). fo) Q.5. (a) Consider the set V = R™ with standard addition and scalar multiplication defined (v=0, for any v € R",r € R, where F = R. Check Whether the set V over F forms a vector space or not? 5 | (b) Find Eigen values and bases for Eigen spaces of A= : Q6. (a)IfA= 5 then diagonalize that matrix (b) Prove that the set { x + iy|x, y are real numpers) forms a Vector Space. LK-9297/19-11-25 -- ustadni.com