# IIT JAM Mathematics Test Series 2023 : Linear Algebra – Linear Transformation and its properties

Practice IIT JAM Mathematics Test Series for Free only On www.examflame.com and take your preparation to the another level. In the Series of Tests for IIT JAM 2023 this is the test of topic Linear Transformation and its properties of the Chapter Linear Algebra. These Questions are prepared as per the Latest Syllabus of IIT JAM Mathematics 2023 . Practicing mock tests/Test series, you will get an idea about how and which type of Question will ask in the Examination. It also boost your confidence level. Also Solve IIT JAM Mathematics Previous Year Question Paper. And Don’t Forget to Shore with Your Friends.

## IIT JAM Mathematics Test Series 2023 : Linear Algebra – Linear Transformation and its properties

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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