Worksheet: Linear Transformations in Planes: Rotation

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In this worksheet, we will practice finding the matrix of linear transformation of rotation at a given angle and the image of a vector under a given rotation linear transformation.

Q1:

Find the matrix for the linear transformation that rotates every vector in ℝ through an angle of 𝜋3.

  • A⎡⎢⎢⎢⎣12√32√3212⎤⎥⎥⎥⎦
  • B⎡⎢⎢⎢⎣12−√32√3212⎤⎥⎥⎥⎦
  • C−100−1
  • D⎡⎢⎢⎢⎣√32−1212√32⎤⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎣√321212√32⎤⎥⎥⎥⎦

Q2:

Find the matrix for the linear transformation that rotates every vector in ℝ through an angle of 𝜋4.

  • A⎡⎢⎢⎢⎣12−√32√3212⎤⎥⎥⎥⎦
  • B−100−1
  • C⎡⎢⎢⎢⎣√22√22√22√22⎤⎥⎥⎥⎦
  • D⎡⎢⎢⎢⎣12√32√3212⎤⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎣√22−√22√22√22⎤⎥⎥⎥⎦

Q3:

Find the matrix for the linear transformation which rotates every vector in ℝ through an angle of 𝜋12.

  • A⎡⎢⎢⎢⎣1−√22√2−√32√3−√221−√22⎤⎥⎥⎥⎦
  • B⎡⎢⎢⎢⎣12−√32√3212⎤⎥⎥⎥⎦
  • C⎡⎢⎢⎢⎣√6+√24√2−√64√6−√24√6+√24⎤⎥⎥⎥⎦
  • D⎡⎢⎢⎢⎣√2−√64√6+√24√6+√24√6−√24⎤⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎣1−√22√3−√22√3−√221−√22⎤⎥⎥⎥⎦

Q4:

Find the matrix for the linear transformation which rotates every vector in ℝ through an angle of 2𝜋3.

  • A⎡⎢⎢⎢⎣−12√32√32−12⎤⎥⎥⎥⎦
  • B⎡⎢⎢⎢⎣12−√32√3212⎤⎥⎥⎥⎦
  • C⎡⎢⎢⎢⎣−√32−12−12√32⎤⎥⎥⎥⎦
  • D⎡⎢⎢⎢⎣−√3212−12√32⎤⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎣−12−√32√32−12⎤⎥⎥⎥⎦

Q5:

Find the matrix for the linear transformation which rotates every vector in ℝ through an angle of 5𝜋12.

  • A⎡⎢⎢⎢⎣√6−√24√6+√24√6+√24√6−√24⎤⎥⎥⎥⎦
  • B⎡⎢⎢⎢⎣−1−√22√2−√32√3−√22−1−√22⎤⎥⎥⎥⎦
  • C⎡⎢⎢⎢⎣√6−√24−√2−√64−√2−√64√2−√64⎤⎥⎥⎥⎦
  • D⎡⎢⎢⎢⎣√6−√24−√2−√64√6+√24√6−√24⎤⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎣−1−√22√3+√22√3−√22−1−√22⎤⎥⎥⎥⎦

Q6:

Find the matrix for the linear transformation which rotates every vector in ℝ through an angle of −𝜋3.

  • A⎡⎢⎢⎢⎣12√32−√3212⎤⎥⎥⎥⎦
  • B⎡⎢⎢⎢⎣12−√32−√3212⎤⎥⎥⎥⎦
  • C⎡⎢⎢⎢⎣√32−12−12√32⎤⎥⎥⎥⎦
  • D⎡⎢⎢⎢⎣√3212−12√32⎤⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎣12−√32√3212⎤⎥⎥⎥⎦

Q7:

Find, with respect to the standard basis, the matrix which rotates every vector in ℝ counterclockwise about the origin through an angle of 𝜋4.

  • A⎡⎢⎢⎢⎣√22−√22√22√22⎤⎥⎥⎥⎦
  • B⎡⎢⎢⎢⎣12−√22√2212⎤⎥⎥⎥⎦
  • C⎡⎢⎢⎢⎣√22√22√22√22⎤⎥⎥⎥⎦
  • D⎡⎢⎢⎢⎣√22−√22−√22√22⎤⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎣−12−√22√2212⎤⎥⎥⎥⎦

Q8:

Find, with respect to the standard basis, the matrix which rotates every vector in ℝ counterclockwise about the origin through an angle of 𝜋3.

  • A⎡⎢⎢⎢⎣12−√32√32−12⎤⎥⎥⎥⎦
  • B⎡⎢⎢⎢⎣12√32√3212⎤⎥⎥⎥⎦
  • C⎡⎢⎢⎢⎣−√3212−12√32⎤⎥⎥⎥⎦
  • D⎡⎢⎢⎢⎣12−√32√3212⎤⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎣−√3212√3212⎤⎥⎥⎥⎦

Q9:

Find, with respect to the standard basis, the matrix which rotates every vector in ℝ counterclockwise about the origin through an angle of 𝜋12.

  • A⎡⎢⎢⎢⎣√6+√24√2−√64√6−√24√6+√24⎤⎥⎥⎥⎦
  • B⎡⎢⎢⎢⎣√6+√24√6+√24√6−√24√2−√64⎤⎥⎥⎥⎦
  • C⎡⎢⎢⎢⎣√6+√24−√6+√24√6−√24−√2−√64⎤⎥⎥⎥⎦
  • D⎡⎢⎢⎢⎣√6+√24−√6+√24√6+√24√2−√64⎤⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎣√6+√24√2−√64√2−√64−√6+√24⎤⎥⎥⎥⎦

Q10:

Find, with respect to the standard basis, the matrix which rotates every vector in ℝ counterclockwise about the origin through an angle of 5𝜋12.

  • A⎡⎢⎢⎢⎣−√6+√24−√6+√24−√6+√24−√6−√24⎤⎥⎥⎥⎦
  • B⎡⎢⎢⎢⎣√6−√24−√6+√24√6+√24−√6−√24⎤⎥⎥⎥⎦
  • C⎡⎢⎢⎢⎣−√6−√24√6+√24√6+√24−√6−√24⎤⎥⎥⎥⎦
  • D⎡⎢⎢⎢⎣√6+√24−√6+√24−√6+√24√6−√24⎤⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎣√6−√24−√6+√24√6+√24√6−√24⎤⎥⎥⎥⎦

Q11:

Describe the geometric effect of the transformation produced by the matrix ⎡⎢⎢⎢⎣√22−√22√22√22⎤⎥⎥⎥⎦.

  • Aa rotation through an angle of 90∘
  • Ba rotation through an angle of 45∘
  • Ca rotation through an angle of −45∘
  • Da rotation through an angle of −90∘
  • Ea rotation through an angle of 135∘

Q12:

A rotation with center the origin sends the vector 34 to 43. Find the matrix representation of this rotation.

  • A⎡⎢⎢⎣453535−45⎤⎥⎥⎦
  • B⎡⎢⎢⎣2425725−7252425⎤⎥⎥⎦
  • C⎡⎢⎢⎣4535−3545⎤⎥⎥⎦
  • D0110
  • E⎡⎢⎢⎣2425725725−2425⎤⎥⎥⎦

Q13:

Let 𝐴 be the matrix of the transformation which rotates all vectors in ℝ through an angle of 𝜃, where 0<𝜃≤2𝜋. For which values of 𝜃 does 𝐴 have a real eigenvalue?

  • A𝜃=0 and 𝜃=𝜋
  • B𝜃=0 and 𝜃=3𝜋2
  • C𝜃=0 and 𝜃=𝜋3
  • D𝜃=𝜋2 and 𝜃=𝜋
  • E𝜃=0 and 𝜃=𝜋2

Q14:

Consider the linear map 𝑇∶ℝ→ℝ, which acts by rotating a vector 𝜃∘ counterclockwise around the origin. Which of the following is the matrix of the linear transformation?

  • A𝜃𝜃𝜃𝜃sincoscossin
  • B𝜃𝜃−𝜃𝜃cossinsincos
  • C−𝜃𝜃−𝜃𝜃cossinsincos
  • D𝜃−𝜃𝜃𝜃sincoscossin
  • E𝜃−𝜃𝜃𝜃cossinsincos

Q15:

Consider the linear transformation 𝐿∶ℝ→ℝ, where 𝐿 rotates each vector 90 degrees about the positive 𝑥-axis. Find the matrix representation of 𝐿 in the standard basis.

  • A10000−10−10
  • B10000−1010
  • C100001010
  • D10010−1110

Q16:

A linear transformation of a plane sends the vector 10 to 𝑝𝑞. If the transformation is a rotation about the origin, where does it send 01?

  • A−𝑞𝑝
  • B𝑝𝑞
  • C−𝑞−𝑝
  • D𝑞−𝑝
  • E𝑞𝑝

Q17:

A linear transformation of a plane sends vector 10 to 𝑝𝑞. If the transformation is a rotation, where does it send 01?

  • A−𝑞−𝑝
  • B𝑞𝑝
  • C𝑝𝑞
  • D−𝑞𝑝
  • E𝑞−𝑝

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