Linear Transform MCQ - 2 - Mathematics MCQ

Let T be a 4 x 4 real matrix such that T⁴ = 0. Let k_i = dim(kerTⁱ) for 1 < i < 4 which of the following is not a possibility for the sequence k₁ < k₂ < k₃ < k₄?

Let  be a linear transformation. Which of the following statement implies that T is bijective?

Let  be linear transformation such that T _° S is the identity map of Then

Let n be a positive integer and let denotes the space of all n x n real matrices. If is a linear transformation such thatT(A) = 0 whenever is symmetric or skew symmetric then the rank of T is

Let A be a 3 × 3 matrix with eigenvalues 1, –1 and 3. Then 

Let S = {T: R³ → R³; T is a linear transformation with T (1, 0, 1) = (1, 2, 3) and T (1, 2, 3) = (1, 0, 1). Then S is 

Let be the linear transformation defined by T(x₁, x₂, x₃) = (x₁ + 3x₂ + 2x₃, 4x₂ + x₃, 2x + x₂ - x₃). The dimension of null space of T³ is

Let the linear transformation be defined by T(x₁, x₂) = (x₁, x₂ + x₂, x₂). Then the nullity of T is

Let T be an arbitrary linear transformation from which is not one one then

Let be the linear transformation defined by T(x₁, x₂, x₃) = (x₁+ 3x₂ + 2x₃, 3x₁ + 4x₂ + x₃, 2x₁ + x₂ - x₃).The dim ension of the null space of T² is

Choose the correct matching from a, b. c and d for the transformation T₁, T₂ and T₃ as defined in group I with the statements given in group II

Let T (x, y, z) = xy² + 2z – x²z² be the temperature at the point (x, y, z). The unit vector in the direction in which the temperature decrease most rapidly at (1, 0, – 1) is

Let T: R⁴ → R⁴ be a linear transformation satisfy T³ + 3T² = 4I, where I is the identity 

 be the linear map defined by T (A) = AM where V be the vector space of all 2 x 2 real matrices. Then rank and nullity of T respectively

Let T be a linear transformation from a vector space into a vector space with U as finite dimensional. The rank of T is the dimension of the

Let a matrix A_mxn represents a linear mapping T : V_n → V_m.Then range ofT is generated by

T is non singular iff

Let T : R² → R² be a linear transformation such that T((1, 2)) = (2, 3) and T((0, 1)) = (1, 4).Then T((5, -4)) is

Let T be a linear operator on ℝ³. Let f(X) ∈ ℝ[X] denote its characteristic polynomial. Consider the following statements.

(a). Suppose T is non-zero and 0 is an eigen value of T. If we write f(X) = X g(X) in ℝ[X], then the linear operator g(T) is zero.

(b). Suppose 0 is an eigenvalue of T with at least two linearly independent eigen vectors. If we write f(X) = X g(X) in ℝ[X], then the linear operator g(T) is zero.

The matrix A is represented asThe transpose of the matrix of this matrix is represented as?

Let  be a linear transformation given by T(x, y, z) = (x, y, 0). Then the null space is generated by which one of the following?

Let  and  be two linear transformation. Then which of the following can be true?

What is the rank of the linear transformation defined by T(x, y, z) = (y, 0, z)?

Consider the linear map   is the set of all real valued continuously differential functions defined by D(f) = f' then

Consider the linear transformation  is given by T(x, y, z, u) = (x, y, 0, 0). Then which one of the following is correct?

Let the linear transform ations S and  be defined by 

S(x, y, z) = (2x, 4x - y, 2x + 3y - z)
T(x, y, z) = (x cosθ - y sinθ, sinθ + y cos θ, z)   where