PracticeVector Space Questions and Answers

**Vector Space Questions and Answers**

**Q1. Let P _{n}(x) = x^{n} for x∈R and let V= span{p_{o},p_{1},p_{2},..}. 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) {p_{o},p_{1},p_{2},..} 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 R ^{3} ?**

(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 = {e _{1},e_{2},…,e_{n}}. Let x_{1},x_{2},…,x_{n} ∈ F . Let**

**C = {x _{1}e_{1},x_{1}e_{1 }+ x_{2}e_{2},…,x_{1}e_{1} + x_{2}e_{2} + … +x_{n}e_{n}}. 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**

**a _{1} = (1,1,0,1,0,0), a_{2} = (1,1,0,0,1,0)**

**a**

_{3}= (1,1,0,0,0,1), a_{4}= (1,0,1,1,0,0)**a**

_{5}= (1,0,1,0,1,0), a_{6}= (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 {v _{1},…,v_{n}}be a linearly independent subset of a vector space V where n≥4. Set w_{ij} = v_{i} – v_{j} . Let W be the span of**

**{w**

_{ij}| 1 ≤ i , j ≤ n }. Then(a) {w_{ij} | 1 ≤ i < j ≤ n } spans W

(b) {w_{ij }| 1 ≤ i < j ≤ n } is a linearly independent subset of W.

(c) {w_{ij} | 1 ≤ i ≤ n-1 , j = i + 1 } spans W.

(d) dimW = n.

**Q7. Let V be a 3-dimensional vector space over the field F _{3} = ^{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 u _{1},u_{2},u_{3},…,u_{n} be n linearly independent elements in a vector space over R** .

**Set u**

_{0}= 0 and u_{n+1}= u_{1}. Define v_{i}= u_{i }+ u_{i+1}and w_{i}= u_{i-1}+ u_{i}for i = 1,2,…,n . Then(a) v_{1},v_{2},…,v_{n} are linearly independent, if n = 2010

(b) v_{1},v_{2},…,v_{n} are linearly independent, if n = 2011

(c) w_{1},w_{2},..,w_{n} are linearly independent, if n = 2010

(d) w_{1},w_{2},…,w_{n} are linearly independent, if n = 2011

**Q9. The dimension of the vector space of all symmetric matrices A = (a _{jk}) of order n×n (n ≥ 2) with real entries. a_{11} = 0 and trace zero is**

(a) (n^{2}+n-4) /2

(b) (n^{2}-n+4) /2

(c) (n^{2}+n-3) /2

(d) (n^{2}-n+3) /2

**Q10. Let V _{1},V_{2} be subspaces of a vector space V . which of the following is necessarily a subspace of V?**

(a) V_{1}∩V_{2}

(b) V_{1}∪V_{2}

(c) V_{1} + V_{2} = {x + y : x∈V_{1}, y∈V_{2}}

(d) V_{1} / V_{2} ={x∈V_{1} and x∉V_{2}}

**Q11. Let S = {A:A[a _{ij}]_{5×5} , a_{ij} = 0 or 1 ∀i, j, Σ_{i} a_{ij} = 1∀j and Σ_{j} a_{ij} = 1∀i }**

**Then the number of elements is S is**

(a) 5^{2}

(b) 5^{5}

(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) (n^{2}-n)/_{2} – 1

(b) (n^{2}-2n)/_{2} – 1

(c) (n^{2}+n)/_{2} – 1

(d) (n^{2}+2n)/_{2} – 1

**Q13. Let V = {(x _{1},…; x_{100}) ∈ R^{100 }| x_{1} = 2x_{2} = 3x_{3} and x_{51} – x_{52} – … -x_{100} = 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 {x _{1}, x_{2}, x_{3}} be a basis for V . Then**

(a) {x_{1}+x_{2}, x_{2}, x_{3}} is a basis for V

(b) The dimension of V is 3

(c) {x_{1}-x_{2}, x_{2}-x_{3}, x_{1}-x_{3}} is a basis for V

(d) None of these

**Q15. Let V be the set of all real n×n matrices A=(a _{ij}) with the property that a_{ij} = -a_{ji} for all i, j = 1, 2,…,n Then**

(a) V is a vector space of dimension n^{2} – n

(b) For every A in V, a_{ij} = 0 for all i = 1,2,…,n

(c) V consists of only diagonal matrices.

(d) V is a vector space of dimension (n^{2}-n)/_{2}

NOTE:- If you need anything else like e-books, video lectures, syllabus, etc. regarding your Preparation / Examination then do 📌 mention it in the Comment Section below. |

*Answer Key*

*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) |