Worksheet: Matrix Differentiation

In this worksheet, we will practice finding the matrix of the linear transformation represented by the differential operator.

Q1:

Consider the vector space of polynomials of degree three at most. The differentiation operator 𝐷 is a linear transformation on this vector space. Find the matrix that represents the linear transformation 𝐿 = 𝐷 + 2 𝐷 + 1 with respect to the basis 1 , 𝑥 , 𝑥 , 𝑥 .

  • A 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1
  • B 1 1 0 0 0 1 2 0 0 0 1 3 0 0 0 1
  • C 1 1 2 0 0 1 1 6 0 0 1 1 0 0 0 1
  • D 1 2 2 0 0 1 4 6 0 0 1 6 0 0 0 1
  • E 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

Q2:

Consider the vector space of polynomials of degree three at most. The differentiation operator 𝐷 is a linear transformation on this vector space. Find the matrix that represents the linear transformation 𝐿 = 𝐷 + 5 𝐷 + 4 with respect to the basis 1 , 𝑥 , 𝑥 , 𝑥 .

  • A 4 1 0 0 0 4 2 0 0 0 4 3 0 0 0 4
  • B 1 1 0 0 0 1 2 0 0 0 1 3 0 0 0 1
  • C 4 1 0 0 0 4 1 0 0 0 0 4 1 5 0 0 0 4
  • D 4 5 2 0 0 4 1 0 6 0 0 4 1 5 0 0 0 4
  • E 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4

Q3:

Consider the vector space of infinitely differentiable functions. The differentiation operator 𝐷 is a linear transformation on this vector space. Find the general form of an element of the kernel of the linear transformation 𝐴 = 𝐷 + 2 𝐷 + 1 2 .

  • A 𝑦 ( 𝑡 ) = 𝐶 𝑡 𝑒 + 𝐶 𝑒 1 𝑡 2 𝑡
  • B 𝑦 ( 𝑡 ) = 𝐶 𝑡 𝑒 1 𝑡
  • C 𝑦 ( 𝑡 ) = 𝐶 𝑡 𝑒 1 𝑡
  • D 𝑦 ( 𝑡 ) = 𝐶 𝑡 𝑒 + 𝐶 𝑒 1 𝑡 2 𝑡
  • E 𝑦 ( 𝑡 ) = 𝐶 𝑡 𝑒 𝐶 𝑒 1 𝑡 2 𝑡

Q4:

Consider the vector space of infinitely differentiable functions. The differentiation operator 𝐷 is a linear transformation on this vector space. Find the general form of an element of the kernel of the linear transformation 𝐴 = 𝐷 + 5 𝐷 + 4 2 .

  • A 𝑦 ( 𝑡 ) = 𝐶 𝑒 + 𝐶 𝑒 1 𝑡 2 4 𝑡
  • B 𝑦 ( 𝑡 ) = 𝐶 𝑒 1 𝑡
  • C 𝑦 ( 𝑡 ) = 𝐶 𝑒 1 𝑡
  • D 𝑦 ( 𝑡 ) = 𝐶 𝑒 + 𝐶 𝑒 1 𝑡 2 4 𝑡
  • E 𝑦 ( 𝑡 ) = 𝐶 𝑒 𝐶 𝑒 1 𝑡 2 4 𝑡

Q5:

Apply the linear differential operator 𝐷 to evaluate the following expression: 𝐷 2 𝐷 + 4 𝑥 𝑒 + 5 𝑥 + 2 2 𝑥 2 .

  • A 𝑥 𝑒 + 2 5 𝑥 𝑥 2
  • B 3 𝑥 𝑒 1 8 + 2 0 𝑥 2 0 𝑥 𝑥 2
  • C 𝑥 𝑒 + 1 0 𝑥 𝑥
  • D 3 𝑥 𝑒 + 1 8 2 0 𝑥 + 2 0 𝑥 𝑥 2

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