Questions on Linear Transformation with Answers

Questions on Linear Transformation with Answers

Questions on Linear Transformation with Answers

Q1. Consider non-zero vector space V1,V2,V3,V4 and linear transformations Φ1 : V1 → V2, Φ2 : V2 → V3, Φ3 : V3 → V4 such that ker(Φ1) = {0}, Range (Φ1) = ker(Φ2), Range(Φ2) = ker(Φ3), Range(Φ3) = V4

(a) i=1Σ4 (-1)i dim Vi = 0
(b) i=1Σ4 (-1)i dim Vi > 0
(c) i=1Σ4 (-1)i dim Vi < 0
(d) i=1Σ4 (-1)i dim Vi ≠ 0

Q2. Let Mn(K) denote the space of all n×n matrices with entries from a field K. Fix a non-singular matrix A = (Aij) ∈ Mn(K) and consider the linear map T: Mn(K) → Mn(K) given by : T(X) = AX. Then

(a) Trace (T) = n i=1Σn Aii
(b) Trace (T) = i=1Σn j=1Σn Aij
(c) Rank of T is n2
(d) T is non-singular

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

(a) There exists A ∈ M2×5(R) such that the dimension of the null space of A is 2
(b) There exists A ∈ M2×5(R) such that the dimension of the null space of A is 0
(c) There exists A ∈ M2×5(R)  and B ∈ M5×2(R) such that AB is the 2×2 identity matrix
(d) There exists A∈ M2×5(R) whose null space is {(x1,x2,x3,x4,x5) ∈ R5 : x1 = x2 = x3 = x4 = x5}

Q4. Let V be the vector space of Polynomials  over R of degree less than or equal to n. For p(x) = a0 + a1x + … + anxn in V, define a linear transformation T : V → V by (Tp)(x) = a0 – a1x + a2x2 – … + (-1)n anxn. 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:Rn → Rn 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 Mn(R) denotes the space of all n×n real matrices . If T: Mn(R) → Mn(R) is a linear transformation such that T(A) = 0 whenever A ∈ Mn(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: R3 → R4 and T:R4 → R3 be linear transformations such that T°S is the identity map of R3. Then

(a) S°T is the identity map of R4
(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 {e1,e2,…,en+1} is a basis of V and T:V → V is the linear transformation satisfying T(ei) = ei+1 for i = 1,2,…,n and T(en+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 Bn = {1,x,x2,…,xn}. If T:P3 → P4 is the linear transformation defined by T[p(x)] = x2p'(x) + 0xp(t) dt and A = [aij] is the 5×4 matrix of T with respect  to standard  bases B3 and B4, then

(a) a32 = 3/2 and a33 = 7/3
(b) a32 = 3/2 and a33 = 0
(c) a32 = 0 and a33 = 7/3
(d) a32 = 0 and a33 = 0

Q10. Consider the linear transformation T:R7 → R7 defined by T(x1,x2,…,x6,x7) = (x7,x6,…,x2,x1) Which of the following are true ?

(a) The determinant of T is 1
(b) There is a basis of R7 with respect to which T is diagonal matrix
(c) T7 = I
(d) The smallest n such that Tn = I is even

Q11. Let V be the vector space of all linear transformations from R3 to R2 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:R6 → R5 and B:R5 → R7 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) : R3 → R2 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:R3 → W be the orthogonal projection of R3 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. {v1,v2,v3} is a basis of V = R3 and a linear transformation T:V → V if defined by T(v1) = v1 + v2, T(v2) = v2 + v3, T(v3) = v3 + v1, then

(a) T is 1-1
(b) T(v1+v2+v3) = 0
(c) T(v1+2v2+v3) = T(v2+2v3)
(d) T(v1-v3) = v1-v3

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