Linear Transform MCQ - 1


30 Questions MCQ Test Topic-wise Tests & Solved Examples for IIT JAM Mathematics | Linear Transform MCQ - 1


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This mock test of Linear Transform MCQ - 1 for Mathematics helps you for every Mathematics entrance exam. This contains 30 Multiple Choice Questions for Mathematics Linear Transform MCQ - 1 (mcq) to study with solutions a complete question bank. The solved questions answers in this Linear Transform MCQ - 1 quiz give you a good mix of easy questions and tough questions. Mathematics students definitely take this Linear Transform MCQ - 1 exercise for a better result in the exam. You can find other Linear Transform MCQ - 1 extra questions, long questions & short questions for Mathematics on EduRev as well by searching above.
QUESTION: 1

The unique linear transformation such that T(1,2) = (2,3) and T(0,1) = (1,4). Then H rule for T is

Solution:
QUESTION: 2

Let V and W be vector spaces over be a map. Then T is a linear transformation iff

Solution:
QUESTION: 3

 is a linear transformation T(1,0) = (2,3,l) and T(1,1) = (3,0,2) then which one of the following statement is correct?

Solution:
QUESTION: 4

Let T:R2 -> R2 be the transformation T(x1,x2) = (x1,0). The null space (or kernel) N(T) of T is

Solution:
QUESTION: 5

defined by differentiation and integration 

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QUESTION: 6

Which of the following is not linear?

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QUESTION: 7

be the vector space of all complex numbers over complex field be defined by T(z) =

Solution:
QUESTION: 8

Given then find a + b.

Solution:

AA-1 = I =
Therefore, a = 1 / 60 and b = 1 / 3 and a + b = 7 / 20.

QUESTION: 9

Let T:  R3 → R3 be a linear transformation and I be the identify transformation of  R3. If there is a scalar C and a non-zero vector x ∈ R3 such that T(x) = Cx, then rank (T – CI) 

Solution:

By rank-nullity is theorem, 
dim(T) = Rank(T) + Nullity (T) 
Here dim(T) = 3 Now, (T – CI)x = T(x) – (T(x) = cx – cx 
= 0
(I is identity transformation) 
⇒ Nullity of T – CI cannot be zero 
⇒ Hence, Rank of T – CI cannot be 3.

QUESTION: 10

Which of the following Linear Transformations is not correct for the given matrix?

Solution:

In the given question,

Thus, x1 = 1y1 - 2y2 - 3y3

x2 = -1y1 + 1y3

x3 = 2y1 + y2.

QUESTION: 11

Find the inverse Fourier transform of

Solution:


Applying inverse Fourier transform, we get
x(t) = -e-2t u(t) + 5e-4t u(t).

QUESTION: 12

Consider the basis S = {v1, v2, v3} for where v1 = (1,1,1) and v2 = (1,1,0), v3 = (1,0,0) and let  be a linear transformation such that T(v1) = (1,0), T (v2) = (2, -1), T (v3) = (4, 3). Then T (2, - 3, 5) is

Solution:
QUESTION: 13

is set of real numbers in let f1 and f2 be the two transformation defined by f1(x,y) = (0,y), f2(x,y) = (y,x). Then the product of the mapping (f2 ° f1)(x,y) gives the projection of the x - y plane on the

Solution:

 

QUESTION: 14

Consider the following  such that T(2,2) = (8, - 6), T (5, 5) = (3, - 2) Then

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QUESTION: 15

Let T be linear transformation on  into itself such that T(1,0) = (1,2) and T (1, 1) = (0, 2) .Then T(a, b) is equal to

Solution:
QUESTION: 16

Which of the following mapping   is not a linear mapping?

Solution:
QUESTION: 17

If is given by T (x, y, z) = (x - y, y + 3z, x + 2y). Then T-1 is

Solution:
QUESTION: 18

Find the sum of the Eigen values of the matrix

Solution:

According to the property of the Eigen values, the sum of the Eigen values of a matrix is its trace that is the sum of the elements of the principal diagonal.
Therefore, the sum of the Eigen values = 3 + 4 + 1 = 8.

QUESTION: 19

Let T: R3 → R3 be a linear transformation and I be the identity transformation of R3. If there is a scalar C and a non-zero vector x ∈ R3 such that T(x) = Cx, then rank (T – CI)

Solution:

By rank-nullity is theorem,
dim(T) = Rank(T) + Nullity (T)
Here dim(T) = 3
Now, (T – CI)x = T(x) – (T(x) = cx – cx = 0 (I is identity transformation)
⇒ Nullity of T – CI cannot be zero
⇒ Hence, Rank of T – CI cannot be 3.
 

QUESTION: 20

where T be the reflection of the points through the line y = -x then the matrix of T with respect to standard basis is

Solution:
QUESTION: 21

Consider the two linear mapsT1 and T2 on V3 defined as T1(x1, x2, x3) = (0, x2, x3) and T2(x1, x2, x3) = (x1, 0,0)

Solution:
QUESTION: 22

Let T be a linear transformation on the vector space defined by T(a, b) = (a, 0) the matrix of T relative to the ordered basis {(1,0), (0,1)} of is

Solution:
QUESTION: 23

Let W be the vector space of all real polynomials of degree atmost 3. Define T : W → W by T(p(x)) = p'(x) where p'(x) is the derivative of P.The matrix of T in the basis {1, x, x2, x3} considered as column vector is given by

Solution:
QUESTION: 24

Let be the polynomial space with basis {1, x, x2} then matrix representation of  

Solution:
QUESTION: 25

Find the fourier transform of F(x) = 1, |x| < a0, otherwise.

Solution:


QUESTION: 26

Let be defined by T(p(x)) = p"(x) + p'(x). Then the matrix representation of T with respect to basis {1, x, x2, x3} and {1, x, x2} of and  respectively is

Solution:
QUESTION: 27

For the standard basis {(1,0,0), (0,1,0), (0,0,1)} of is a linear transformation T from has the matrix representation Then the image of (2,1,2) under T is

Solution:
QUESTION: 28

Let us consider a 3×3 matrix A with Eigen values of λ1, λ2, λ3 and the Eigen values of A-1 are?

Solution:

According to the property of the Eigen values, if is the Eigen value of A, then 1 / λ is the Eigen value of A-1. So the Eigen values of A-1 are 1 / λ1, 1 / λ2, 1 / λ3.

QUESTION: 29

Let  be the map given by  If the matrix of T relative to the standard basis β = γ = {1, x, x2, x3} is

Solution:
QUESTION: 30

A linear transformation T rotates each vector in clockwise through 90°. The matrix T relative to standard ordered basis 

Solution:

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