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Worksheet: 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 2 with respect to the basis  1 , π‘₯ , π‘₯ , π‘₯  2 3 .

  • A ⎑ ⎒ ⎒ ⎣ 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 ⎀ βŽ₯ βŽ₯ ⎦
  • B ⎑ ⎒ ⎒ ⎣ 1 1 0 0 0 1 2 0 0 0 1 3 0 0 0 1 ⎀ βŽ₯ βŽ₯ ⎦
  • C ⎑ ⎒ ⎒ ⎣ 1 1 2 0 0 1 1 6 0 0 1 1 0 0 0 1 ⎀ βŽ₯ βŽ₯ ⎦
  • D ⎑ ⎒ ⎒ ⎣ 1 2 2 0 0 1 4 6 0 0 1 6 0 0 0 1 ⎀ βŽ₯ βŽ₯ ⎦
  • E ⎑ ⎒ ⎒ ⎣ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ⎀ βŽ₯ βŽ₯ ⎦

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 2 with respect to the basis  1 , π‘₯ , π‘₯ , π‘₯  2 3 .

  • A ⎑ ⎒ ⎒ ⎣ 4 1 0 0 0 4 2 0 0 0 4 3 0 0 0 4 ⎀ βŽ₯ βŽ₯ ⎦
  • B ⎑ ⎒ ⎒ ⎣ 1 1 0 0 0 1 2 0 0 0 1 3 0 0 0 1 ⎀ βŽ₯ βŽ₯ ⎦
  • C ⎑ ⎒ ⎒ ⎣ 4 1 0 0 0 4 1 0 0 0 0 4 1 5 0 0 0 4 ⎀ βŽ₯ βŽ₯ ⎦
  • D ⎑ ⎒ ⎒ ⎣ 4 5 2 0 0 4 1 0 6 0 0 4 1 5 0 0 0 4 ⎀ βŽ₯ βŽ₯ ⎦
  • E ⎑ ⎒ ⎒ ⎣ 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4 ⎀ βŽ₯ βŽ₯ ⎦

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 ( 𝑑 ) = 𝐢 𝑑 𝑒 + 𝐢 𝑒 1 βˆ’ 𝑑 2 βˆ’ 𝑑
  • B 𝑦 ( 𝑑 ) = 𝐢 𝑑 𝑒 1 𝑑
  • C 𝑦 ( 𝑑 ) = 𝐢 𝑑 𝑒 1 βˆ’ 𝑑
  • D 𝑦 ( 𝑑 ) = 𝐢 𝑑 𝑒 + 𝐢 𝑒 1 𝑑 2 𝑑
  • E 𝑦 ( 𝑑 ) = 𝐢 𝑑 𝑒 βˆ’ 𝐢 𝑒 1 𝑑 2 𝑑

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 ( 𝑑 ) = 𝐢 𝑒 + 𝐢 𝑒 1 𝑑 2 4 𝑑
  • B 𝑦 ( 𝑑 ) = 𝐢 𝑒 1 βˆ’ 𝑑
  • C 𝑦 ( 𝑑 ) = 𝐢 𝑒 1 𝑑
  • D 𝑦 ( 𝑑 ) = 𝐢 𝑒 + 𝐢 𝑒 1 βˆ’ 𝑑 2 βˆ’ 4 𝑑
  • E 𝑦 ( 𝑑 ) = 𝐢 𝑒 βˆ’ 𝐢 𝑒 1 𝑑 2 4 𝑑

Q5:

Apply the linear differential operator 𝐷 to evaluate the following expression: ο€Ή 𝐷 βˆ’ 2 𝐷 + 4  ο€Ή π‘₯ 𝑒 + 5 π‘₯ + 2  2 π‘₯ 2 .

  • A π‘₯ 𝑒 + 2 5 π‘₯ π‘₯ 2
  • B 3 π‘₯ 𝑒 βˆ’ 1 8 + 2 0 π‘₯ βˆ’ 2 0 π‘₯ π‘₯ 2
  • C π‘₯ 𝑒 + 1 0 π‘₯ π‘₯
  • D 3 π‘₯ 𝑒 + 1 8 βˆ’ 2 0 π‘₯ + 2 0 π‘₯ π‘₯ 2