Worksheet: Linear Transformation Definition

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In this worksheet, we will practice defining a linear transformation and stating whether a given transformation is linear.

Q1:

Suppose 𝑉 is a finite dimensional vector space and 𝑇 is a linear operator on 𝑉. Then which of the following conditions must be true for a subspace 𝑈⊆𝑉 to be an invariant subspace under 𝑇?

  • A 𝑇 𝑢 ∈ 𝑉 for all 𝑢∈𝑉
  • B 𝑇 𝑢 ∈ 𝑈 for all 𝑢∈𝑈
  • C 𝑇 𝑢 ∈ 𝑉 for all 𝑢∈𝑈
  • D 𝑇 𝑢 ∈ 𝑈 for all 𝑢∈𝑉

Q2:

Consider the linear transformations for which v, the image of 10, and w, the image of 01, are unit vectors. Let 𝐿 be a linear transformation of this kind which has the additional property that the area of the parallelogram with vertices 0, v, w, and vw+ is as big as possible. What are the possible values of the measure of the angle between v and w for the transformation 𝐿?

  • A 0 ∘
  • B 9 0 ∘ only
  • C 1 8 0 ∘
  • D 2 7 0 ∘ only
  • E 9 0 ∘ and 270∘

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