Questions on Linear Transformation with Answers

**Questions on Linear Transformation with Answers**

**Q1. Consider non-zero vector space V _{1},V_{2},V_{3},V_{4} and linear transformations Φ_{1} : V_{1} → V_{2}, Φ_{2} : V_{2} → V_{3}, Φ_{3} : V_{3} → V_{4} such that ker(Φ_{1}) = {0}, Range (Φ_{1}) = ker(Φ_{2}), Range(Φ_{2}) = ker(Φ_{3}), Range(Φ_{3}) = V_{4}**

(a) _{i=1}Σ^{4} (-1)^{i} dim V_{i} = 0

(b) _{i=1}Σ^{4} (-1)^{i} dim V_{i} > 0

(c) _{i=1}Σ^{4} (-1)^{i} dim V_{i} < 0

(d) _{i=1}Σ4 (-1)^{i} dim V_{i} ≠ 0

**Q2. Let M _{n}(K) denote the space of all n×n matrices with entries from a field K. Fix a non-singular matrix A = (A_{ij}) ∈ M_{n}(K) and consider the linear map T: M_{n}(K) → M_{n}(K) given by : T(X) = AX. Then**

(a) Trace (T) = n _{i=1}Σ^{n} A_{ii}

(b) Trace (T) = _{i=1}Σ^{n} _{j=1}Σ^{n} A_{ij}

(c) Rank of T is n^{2}

(d) T is non-singular

**Q3. Let M _{m×n} (R) be the set of all m×n matrices with real entries. Which of the following statement is correct ?**

(a) There exists A ∈ M_{2×5}(R) such that the dimension of the null space of A is 2

(b) There exists A ∈ M_{2×5}(R) such that the dimension of the null space of A is 0

(c) There exists A ∈ M_{2×5}(R) and B ∈ M_{5×2}(R) such that AB is the 2×2 identity matrix

(d) There exists A∈ M_{2×5}(R) whose null space is {(x1,x_{2},x_{3},x_{4},x_{5}) ∈ R^{5} : x_{1} = x_{2} = x_{3 }= x_{4} = x_{5}}

**Q4. Let V be the vector space of Polynomials over R of degree less than or equal to n. For p(x) = a _{0} + a_{1}x + … + a_{n}x^{n} in V, define a linear transformation T : V → V by (T_{p})(x) = a_{0} – a_{1}x + a_{2}x^{2} – … + (-1)^{n} a_{n}x^{n}. Then which of the following are correct ?**

(a) T is one-to-one

(b) T is onto

(c) T is invertible

(d) det T = 0

**Q5. Let T:R ^{n} → R^{n} be a linear transformation. Which of the following statements implies that T is bijective ?**

(a) Nullity (T) = n

(b) Rank (T) = Nullity (T) = n

(c) Rank (T) + Nullity (T) = n

(d) Rank (T) – Nullity (T) = n

**Q6. Let n be a positive integer and let M _{n}(R) denotes the space of all n×n real matrices . If T: M_{n}(R) → M_{n}(R) is a linear transformation such that T(A) = 0 whenever A ∈ M_{n}(R) is symmetric or skew-symmetric, then the rank of T is**

(a) ^{n(n+1)}/_{2}

(b) ^{n(n-1)}/_{2}

(c) n

(d) 0

**Q7. Let S: R ^{3} → R^{4} and T:R^{4} → R^{3} be linear transformations such that T°S is the identity map of R^{3}. Then**

(a) S°T is the identity map of R^{4}

(b) S°T is one-one, but not onto

(c) S°T is onto, but not one-one

(d) S°T is neither one-one or onto.

**Q8. Let n be a positive integer and V be an (n+1) – dimensional vector space over R. If {e _{1},e_{2},…,e_{n+1}} is a basis of V and T:V → V is the linear transformation satisfying T(e_{i}) = e_{i}+1 for i = 1,2,…,n and T(e_{n}+1) = 0. Then**

(a) Trace of T is non zero

(b) Rank of T is n

(c) Nullity of T is 1

(d) Tn = T•T•…•T (n times) is the zero map.

**Q9. For a positive integer n, let Pn denote the space of all Polynomials p(x) with coefficients in R such that deg p(x) ≤ n, and let B _{n} = {1,x,x^{2},…,x^{n}}. If T:P^{3} → P^{4} is the linear transformation defined by T[p(x)] = x^{2}p'(x) + _{0}∫^{x}p(t) dt and A = [a_{ij}] is the 5×4 matrix of T with respect to standard bases B_{3} and B_{4}, then**

(a) a_{32} = ^{3}/_{2} and a_{33} = ^{7}/_{3}

(b) a_{32} = ^{3}/_{2} and a_{33} = 0

(c) a_{32} = 0 and a_{33} = ^{7}/_{3}

(d) a_{32} = 0 and a_{33} = 0

**Q10. Consider the linear transformation T:R ^{7} → R^{7} defined by T(x_{1},x_{2},…,x_{6},x_{7}) = (x_{7},x_{6},…,x_{2},x_{1}) Which of the following are true ?**

(a) The determinant of T is 1

(b) There is a basis of R^{7} with respect to which T is diagonal matrix

(c) T^{7} = I

(d) The smallest n such that T^{n} = I is even

**Q11. Let V be the vector space of all linear transformations from R ^{3} to R^{2} under usual addition and scalar multiplication. Then**

(a) V is a vector space of dimension 5

(b) V is a vector space of dimension 6

(c) V is a vector of dimension 8

(d) V is a vector space of dimension 9

**Q12. Let A:R ^{6} → R^{5} and B:R^{5} → R^{7} be two linear tansformations. Then which of the following can be true ?**

(a) A and B are none one-one

(b) A is one-one and B is not one-one

(c) A is onto and B is one-one

(d) A and B both are onto

**Q13. The transformation (x,y,z) → (x+y, y+z) : R ^{3} → R^{2} is**

(a) Linear and has zero kernel

(b) Linear and has a proper subspace as kernel

(c) Neither linear nor 1-1

(d) Neither linear nor onto

**Q14. Let T:R ^{3} → W be the orthogonal projection of R^{3} on to the xz – plane W. Then**

(a) T(x,y,z) = (x+y, 0, y+z)

(b) T(x,y,z) = (x-y, 0, y-z)

(c) T(x,y,z) = (x+y+z, 0, z)

(d) T(x,y,z) = (x, 0, z)

**Q15. {v _{1},v_{2},v_{3}} is a basis of V = R_{3} and a linear transformation T:V → V if defined by T(v_{1}) = v_{1} + v_{2}, T(v_{2}) = v_{2} + v_{3}, T(v_{3}) = v_{3} + v_{1}, then**

(a) T is 1-1

(b) T(v_{1}+v_{2}+v_{3}) = 0

(c) T(v_{1}+2v_{2}+v_{3}) = T(v_{2}+2v_{3})

(d) T(v_{1}-v_{3}) = v_{1}-v_{3}

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*Answer Key*

*Answer Key*

01. | (a), (b) | 06. | (d) | 11. | (b) |

02. | (a), (c), (d) | 07. | (d) | 12. | (a), (c) |

03. | (c) | 08. | (b), (c) | 13. | (b) |

04. | (a), (b), (c) | 09. | (b) | 14. | (d) |

05. | (d) | 10. | (b), (d) | 15. | (a) |