Subject Code: Subject Name: SEM/ Branch: Faculty In-charge:
ECC 401 Engineering Mathematics-IV IV/ EXTC Zeba Ansari
QB TUT, IAT2 _Vector
Space Course Outcome: CO4
Question CO BTL
No. level Question
1 4 3 Find a vector orthogonal to both u= ( -6, 4,2 ), v= ( 3,1,5 ) May 24,18,Dec18
2 4 3 Find a uint vector orthogonal to both u= ( -3, 2,1 ), v= ( 3,1,5 ) Dec17
3 4 3 If the ( k, k,-2 ), ( k,-2,12 ) are orthogonal. Find the value of k. Dec21
4 4 3 Determine whether the given vectors (-4,6,-10,1),(2,1,-2,9) are Dec19
orthogonal with respect to Eucildean inner product.
5 4 3 Verify Cauchy Schwartz inequality for the following
(i) u = (-4,2,1), v=(8,-4,-2) May24
(ii) u = (2,1,-3), v=(3,4,-2) Dec24
(iii) u = (1,2,4), v=(-3,2,5) Dec22
(iv) u = (-4,2,1), v=(8,-4,-2) Dec19
(v) u = (-4,2,1 ), v=(8,-4,-2) also find angle between them.
May18
6 4 3 For real values of a , b , θ , Using Cauchy Schwartz inequality, show Dec21
that ( a cosθ +b sinθ )2 ≤ a2 +b 2
7 4 3 Construct an orthonormal basis of R3 using Gram Schmidt Process to
(i) S={( 3 , 0 , 4 ) , (−1 , 0 ,7 ) , ( 2 , 9 , 11) } May 24
(ii) S={( 1 ,2 , 0 ) , ( 0 , 3 ,1 ) } Dec24
(iii) S={( 1 ,1 , 1 ) , (−1 , 1 , 0 ) , ( 1 , 2 ,1 ) } Dec22
(iv) S={( 1 , 0 ,0 ) , ( 3 , 7 ,−2 ) , ( 0 , 4 ,1 ) } May19, Dec19, May18
(v) S={( 1 , 0 ,1 , 1 ) , (−1 , 0 ,−1 , 1 ) , ( 0 ,−1 ,1 , 1 ) } Dec18
(vi) S={( 3 , 0 , 4 ) , (−1 , 0 ,7 ) , ( 2 , 9 , 11) } May17
(vii) S={( 1 ,2 , 0 ) , ( 0 , 3 ,1 ) } Dec15
(viii)
8 4 3 Using Gram Schmidt Process construct an orthonormal basis for the plane Dec21
x+y+z=0.
9 4 3 Determine whether the vectors of the form (a,b,c) where b=a+c form May24
a subspace of R3 under usual addition and scalar multiplication.
10 4 3 Let W be the set of 2 X 2 matrices of the form [ a 0 0 b ] where a and b Dec22
are real numbers. Show that W is a subspace of vector space V of all
2 X 2 matrices.
11 4 3 Check whether the following are subspaces of R3 . May19
(i) W= { ( a,0,0,) / a ϵ R
(ii) W = { ( x,y,z) / x=1 , y=1, z=1 , y ϵ R }
12 4 3 Check whether the set of all pairs of real numbers of the form (1,x) May18
with operations
(1,a) + (1,b) = ( 1, a + b ) and k ( 1,a) = ( 1, k. a ) is a vector space ,
where k is real number.
13 4 3 Check whether the set of real number ( x, 0 ) with operations May17
(x1, 0)+(x2,0) = (x1 x2, 0) , k (x1,0) =( k x1,0) is a vector space.
14 4 3 Prove that W = { (x,y)/ x = 3y } subspace of R2. Is W1 = { ( a,1,1/ a in R)} May16
subspace of R3 ?
15 4 3 Chek whether W = { ( x, y, z ) / y = x+z , x, y, z are in R} is a subspace of R3 Dec16
with usual addition and usual multiplication.
n
16 4 3 If W = { α /αϵ R ∧a1 >0 } a subspace of V= Rn with Dec15
α =¿ in R ( n≥ 1 ). Show that W is not a subspace of V by giving suitable
n
counter example.
