Space Science BS University Of Sargodha (UOS) 2025

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_ £5 3 3 (hath or 38 — x A EN Tr $10 or ’S ees ke) CATE, P Dae wy i Sgn

compulsory. Attempt any three questions from subjective part. TERN

La ERR ha

ae BE,

2) te short answers of the following in 2-3 lines cg your answer sheet. (B80) = 5T

~~ \)F i Find the value of 7 a Where A = [1,r,1] B=[-2 2 s] |

Ag it. Define Symmetric Matrix 2 WW .

208 i iit. Prove that an inverse of a magix, f it exists is unique. / cO

bs x

iv. Ifds= 5 2] then find (AT)? hg

= v. Find unit vector in the direction of v = (3,4) R=

3

vi. Define orthogonal sub-space.

= AE SER | H

oe IfA = [4 1+ ] then find A

y

viii. Define Diagonal Matrix 5

ix. For what value of cu = (1,-2,¢) and v = (2,1, —1) are orthogonal

X. Define Similar Matrix

| 1 3 [14

xi. What combination of ¢ 5] +d 3] produces 3 |

(Compulsory)

Seni lines sin your answer sheet.

valu of and so hat AB] reA= [1711] B=[-2 2 A

Define Symmetric Matrix

. Prove that an inverse of a if it exists is unique.

owas=fl ri Wen find (AT) @® :

v. Find unit vector in the direction of v = (3,4) Ne

vi. Define “tit sub-space.

ws H

vii. IfA= fi 141i ] then find A

viii. Define Diagonal Matrix 5

ix. Por what value of cu = (1,-2,¢) and v = (2,1, —1) are orthogonal

xi. What combination of ¢ I5 ] +d ] produces nl

xii. If A is invertible and AB = AC prove that B = C

cg BR 4 “Subjective Part Cad

Q2 JREETe 3 at Bucs do BR

2. 8) A= i +31 5 ] Find the eig Values and KE ~ ctors bE A and show that cigenvectors are

orthogonal. l oo

2 -1 0 ¥ i

BY fu = - | VE | 2 | # = i] = o findc,d,e suchthatcu + dv+ew =b

0 —1 2 0

Q3.

a)lfS = {u,,u,, u;Phasigorr® where u, = (1,1,1),u, = (—1,0,-1),u; = (—1,2,3)

Use the Gram-Schmidth process to transform S to an orthonormal basis for R?

b) Solve

Hl 1

OE, | o = | by elimination method

BS 7 6

Q.4.

a) Find the volume of box with sides u = (2,4,0),v = (—=1,3,0),w = (1,2,2)

1 0 6]

b)if A= a cul -|o find vector £, Projection Matrix and projection onto b

x29 Wo gl a - \

4 ~ yh

QB) if 4 = - 2 | find 4™" by Cofactor Method

0. ~1 2 yA oi

b) Find v =3t>+5t—5 asa linear; mbination of cP

Po=t*+2t+1, P,=2t2+5t+4,P,=t2+3t +6 Ae

Q.6.

a) W be the subspace of R® spanned by che

u; = (1,2,-1,3,4),u, = (2,4, ~2,68)u; = (13.2.2,6).u, = (145,18) 40 = (27,339)

find the subset of vector that form basis of W and dim. W

b) If v, = (1,2,1),v; = (1,0,2),v; = (1,1,0) than find vector v = (2,1,5) as a linear Combination

of vy, v,,v,

LK-6607/04-06-25

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