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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.
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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:R^{2} -> R^{2} be the transformation T(x_{1},x_{2}) = (x_{1},0). The null space (or kernel) N(T) of T is

Solution:

QUESTION: 5

defined by differentiation and integration

Solution:

QUESTION: 6

Which of the following is not linear?

Solution:

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:

A^{A-1 }= I =

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

QUESTION: 9

Let T: R^{3} → R^{3} be a linear transformation and I be the identify transformation of R3. If there is a scalar C and a non-zero vector x ∈ R^{3} 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, x_{1} = 1y_{1} - 2y_{2} - 3y_{3}

x_{2} = -1y_{1} + 1y_{3}

x_{3 }= 2y_{1} + y_{2}.

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 = {v_{1}, v_{2}, v_{3}} for where v_{1} = (1,1,1) and v_{2} = (1,1,0), v_{3} = (1,0,0) and let be a linear transformation such that T(v_{1}) = (1,0), T (v_{2}) = (2, -1), T (v_{3}) = (4, 3). Then T (2, - 3, 5) is

Solution:

QUESTION: 13

is set of real numbers in let f_{1} and f_{2} be the two transformation defined by f_{1}(x,y) = (0,y), f_{2}(x,y) = (y,x). Then the product of the mapping (f_{2} _{°} f_{1})(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

Solution:

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: R ^{3} → R^{3}**

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 mapsT_{1} and T_{2} on V_{3} defined as T_{1}(x_{1}, x_{2}, x_{3}) = (0, x_{2}, x_{3}) and T_{2}(x_{1}, x_{2}, x_{3}) = (x_{1}, 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, x^{2}, x^{3}} considered as column vector is given by

Solution:

QUESTION: 24

Let be the polynomial space with basis {1, x, x^{2}} 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, x^{2}, x^{3}} and {1, x, x^{2}} 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 λ_{1}, λ_{2}, λ_{3 }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 / λ_{1}, 1 / λ_{2}, 1 / λ_{3}.

QUESTION: 29

Let be the map given by If the matrix of T relative to the standard basis β = γ = {1, x, x^{2}, x^{3}} 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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