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Lesson: Finding the Matrix of the Linear Transformation of Rotating Vectors through a Given Angle

In this lesson, we will learn how to find the matrix of a linear transformation which rotates every vector in R² by a given angle.

Worksheet • 16 Questions

Q1:

Consider the linear transformation 𝐿 β€² ∢ ℝ β†’ ℝ 3 3 , which rotates each vector 45 degrees counterclockwise about the positive π‘₯ -axis then 90 degrees counterclockwise about the positive 𝑦 -axis. Find in the standard basis.

  • A ⎑ ⎒ ⎒ ⎒ ⎒ ⎒ ⎣ 1 √ 2 + √ 2 1 βˆ’ 1 √ 2 + √ 2 ⎀ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ ⎦
  • B  1 2 βˆ’ 1 
  • C ⎑ ⎒ ⎒ ⎒ ⎒ ⎒ ⎣ βˆ’ 1 √ 2 + √ 2 βˆ’ 1 √ 2 βˆ’ √ 2 1 ⎀ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ ⎦
  • D ⎑ ⎒ ⎒ ⎒ ⎣ βˆ’ 1 √ 2 + √ 2 2 βˆ’ 1 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦

Q2:

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

  • A
  • B
  • C
  • D
  • E

Q3:

Find the matrix for the linear transformation which rotates every vector in ℝ 2 through an angle of 5 πœ‹ 1 2 .

  • A ⎑ ⎒ ⎒ ⎒ ⎣ √ 6 βˆ’ √ 2 4 βˆ’ √ 2 βˆ’ √ 6 4 √ 6 + √ 2 4 √ 6 βˆ’ √ 2 4 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • B ⎑ ⎒ ⎒ ⎒ ⎣ √ 6 βˆ’ √ 2 4 βˆ’ √ 2 βˆ’ √ 6 4 βˆ’ √ 2 βˆ’ √ 6 4 √ 2 βˆ’ √ 6 4 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • C ⎑ ⎒ ⎒ ⎒ ⎣ √ 6 βˆ’ √ 2 4 √ 6 + √ 2 4 √ 6 + √ 2 4 √ 6 βˆ’ √ 2 4 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • D ⎑ ⎒ ⎒ ⎒ ⎣ βˆ’ 1 βˆ’ √ 2 2 √ 3 + √ 2 2 √ 3 βˆ’ √ 2 2 βˆ’ 1 βˆ’ √ 2 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • E ⎑ ⎒ ⎒ ⎒ ⎣ βˆ’ 1 βˆ’ √ 2 2 √ 2 βˆ’ √ 3 2 √ 3 βˆ’ √ 2 2 βˆ’ 1 βˆ’ √ 2 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
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