Vector Space Questions and Answers

PracticeVector Space Questions and Answers

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Vector Space Questions and Answers


Q1. Let Pn(x) = xn for x∈R and let V= span{po,p1,p2,..}. Then

(a) V is the vector space of al real valued continuous function on R.
(b) V is a subspace of all real valued continuous function on R.
(c) {po,p1,p2,..} is a linearly independent set in the vector space of all continuous functions on R.
(d) Trigonometric functions belong to V.


Q2. Which of the following are subspaces of the vector space R3 ?

(a) {(x,y,z): x + y = 0 }
(b) {(x,y,z): x – y = 1 }
(c) {(x,y,z): x + y = 0 }
(d) {(x,y,z): x – y = 1 }


Q3. For arbitrary not null subspaces U,V and W of a finite dimensional vector space, which of the following hold:

(a) U ∩ (V+W) ⊂ U∩V + U∩W
(b) U ∩ (V+W) ⊃ U∩V + U∩W
(c) (U∩V) +W) ⊂ (U+W) ∩ (V+W)
(d) (U∩V) +W) ⊃ (U+W) ∩ (V+W)


Q4. Let V denote a vector space over a field F and with a basis B = {e1,e2,…,en}. Let x1,x2,…,xn ∈ F . Let

C = {x1e1,x1e1 + x2e2,…,x1e1 + x2e2 + … +xnen}. Then

(a) C is a linearly independent set implies that xi ≠ 0  for every i = 1,2,…,n
(b) x ≠ 0 for every i = 1,2,…,n implies that C is a linearly independent set.
(c) The linear span of C is V implies that x ≠ 0 for every i =  1,2,…,n
(d) x ≠ 0 for every i = 1,2,…,n implies that the linear span of C is V


Q5. Consider the following row vectors

a1 = (1,1,0,1,0,0),          a2 = (1,1,0,0,1,0)
a3 = (1,1,0,0,0,1),          a4 = (1,0,1,1,0,0)
a5 = (1,0,1,0,1,0),          a6 = (1,0,1,0,0,1)

The dimension of the vector space spanned by these row vectors is

(a) 6
(b) 5
(c) 4
(d) 3


Q6. Let {v1,…,vn}be a linearly independent subset of a vector space V where n≥4. Set wij = vi – vj . Let W be the span of {wij | 1 ≤ i , j ≤ n }. Then

(a) {wij | 1 ≤ i < j ≤ n } spans W
(b) {wij | 1 ≤ i < j ≤ n } is a linearly independent subset of W.
(c) {wij | 1 ≤ i ≤ n-1 , j = i + 1 } spans W.
(d) dimW = n.


Q7. Let V be a 3-dimensional vector space over the field F3 = Z/3Z  of 3 elements. The number of distinct 1-dimensional subspaces of V is

(a) 13
(b) 26
(c) 9
(d) 15


Q8. Let n be an integer, n ≥ 3, and let u1,u2,u3,…,un be n linearly independent elements in a vector space over R . Set u0 = 0  and un+1 = u1 . Define vi = ui + ui+1 and wi = ui-1 + ui for i = 1,2,…,n . Then

(a) v1,v2,…,vn are linearly independent, if n = 2010
(b) v1,v2,…,vn are linearly independent, if n = 2011
(c) w1,w2,..,wn are linearly independent, if n = 2010
(d) w1,w2,…,wn are linearly independent, if n = 2011


Q9. The dimension of the vector space of all symmetric matrices A = (ajk) of order n×n (n ≥ 2) with real entries. a11 = 0 and trace zero is

(a) (n2+n-4) /2
(b) (n2-n+4) /2
(c) (n2+n-3) /2
(d) (n2-n+3) /2


Q10. Let V1,V2 be subspaces of a vector space V . which of the following is necessarily a subspace of V?

(a) V1∩V2
(b) V1∪V2
(c) V1 + V2  = {x + y : x∈V1, y∈V2}
(d) V1 / V2 ={x∈V1 and x∉V2}


Q11. Let   S = {A:A[aij]5×5 , aij = 0 or 1 ∀i, j, Σi aij = 1∀j and Σj aij = 1∀i }

Then the number of elements is S is

(a) 52
(b) 55
(c) 5!
(d) 55


Q12. The dimension of the vector space of all symmetric matrices of order n×n (n 2) with real entries and trace equal to zero is

(a) (n2-n)/2 – 1
(b) (n2-2n)/2 – 1
(c) (n2+n)/2 – 1
(d) (n2+2n)/2 – 1


Q13. Let V = {(x1,…; x100) R100 | x1 = 2x2 = 3x3 and x51 – x52 – … -x100 = 0} . Then

(a) dim V = 98
(b) dim V = 49
(c) dim V = 99
(d) dim V = 97


Q14. Let V be a real vector space and let {x1, x2, x3} be a basis for V . Then

(a) {x1+x2, x2, x3} is a basis for V
(b) The dimension of V is 3
(c) {x1-x2, x2-x3, x1-x3} is a basis for V
(d) None of these


Q15. Let V be the set of all real n×n matrices A=(aij) with the property that aij = -aji for all i, j = 1, 2,…,n Then

(a) V is a vector space of dimension n2 – n
(b) For every A in V, aij = 0 for all i = 1,2,…,n
(c) V consists of only diagonal matrices.
(d) V is a vector space of dimension  (n2-n)/2


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Answer Key

01.(b), (c)06.(a), (c)11.(c)
02.(a), (b)07.(a)12.(c)
03.(b), (b)08.(b), (c), (d)13.(d)
04.(a), (b), (c), (d)09.(a)14.(a), (b)
05.(c)10.(a), (c)15.(d)

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