Linear Algebra/Linear Algebra (LA) BS 3 Semester/Term University Of Sargodha (UOS) 2024

University of Sargodha

BS 3 Term Examination 2024

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.

Oo Objective Part (Compulsory)

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Quin (Wire short answers of the following in 2-3 lines caeh on your answer sheet. (2*12)

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i. Find inverse of A = 2 2) co

4 4 5 AQ

ii. Define Eigen value. xD p

iii. Whether the vectors u; =((§,2, =3),u, = (1,—4,3), are orthogonal or not. p aol

iv. Consider vectors in Ru = (1,1,1),v = (1,2,-3) and w = (1,—4,3) thgneyetivectors are

orthogonal, x AO"

v. Write the bases for the vector space M;,; of 2 X 2 matrices. Nth

vi. Let V be vector space and u € V then show that (=u = —u.

vii. If Ais invertible matrix then A” is also invertible and (A7)~' = (A™1)".

viii. Define Rank and Nullity of homomorphism.

3 ¥ Xieky SN 2Z + CIN IZ ENT.

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ix. Find x,y, z, t such that [cise =a = [5 all

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,K0 =

0.

: — mie ALA Niles vidth trans vara te citbhenace af vector endee of all n % 1 matrices.

Paper: i b TH-3215/MATH-203/MATH-201

oOo Objective Part (Compulsory)

gale short answers of the following] in 2-3 lines AS on your answer sheet, (2*12)

Find inverse of A = \ co

ii. Define Eigen value. * x00

il. Whether the vectors u, =\(£:2, —3),u; = (1,-4.3), are orthogonal or not. ATO

iv. Consider vectors in Ru = (1,1,1) ,v = (1,2,—3) and w = (1,—4,3) ther shel vectors are

orthogonal, . AO

v. Write the bases for the vector space M,,, of 2 X 2 matrices. N=

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

vii. If Ais invertible matrix then A” is also invertible and (A7)~* = (A~ 1)".

Xi y SV 2Z A= CI NZ INT

xX—y z- dq ~l1 sr

x. Define trace of a matrix,

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

ix. Find x,y, z, t such that

Subjective Part (3*12)

0.2.

a) Determine whether (1,1,1,1),(4,2,3,2), (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=; 2] satisty its characteristic equation.

Q.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),u; = (1,2,4,5),u; = (1,-3,-4,-2)

b) Find Eigen values and corresponding Eigen vectors of A = 5 2)

Q.4.

a) Consider the vectors uy = (1,2,1,3,2),u; = (1,3,3,5,3),u, = (3,8,7,13,8), w; = (1,4,69,7),

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

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

X+y+2z=9

~o" 2x+4y-3z=1

Pn) oR 3x+6y-5z=

A “0.5.

a) Let W be subspace of R® spanned by t su, = (1,2,-1,3),u; = (241,-2),

uy = (3,6,3,-7), uy. = (1,2, =A Or s = (2,4,—5,14). Find basis and dimension of W.

2 0 3 _0\

b) Find A", if A=| 0 33 COV

2 0 -4 AO

3 SW

Q.6.

a) lfA= [a 2] then diagonalize that matrix,

b) Let v, =(l, 2 1), v, =(2,9,0)and v, =(3,3,4) . Show that the set S={v,,v,,v,} is basis for rR? ;

LK-6607/14-05-24 )

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