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Lesson: Finding the Matrix of the Linear Transformation of Rotating and Reflecting Vectors

Worksheet • 12 Questions

Q1:

Find the matrix with respect to the standard basis vectors for the linear transformation that rotates every vector in ℝ 2 through an angle of πœ‹ 3 and then reflects it across the 𝑦 -axis.

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

Q2:

A linear transformation is formed by reflecting every vector in in the -axis and then rotating the resulting vector through an angle of . Find the matrix of this linear transformation.

  • A
  • B
  • C
  • D
  • E

Q3:

Let the matrix 𝐴 represent rotation in the plane through an angle of πœƒ and let the matrix 𝐡 represent reflection in the π‘₯ -axis.

What is the matrix 𝐴 ?

  • A  πœƒ βˆ’ πœƒ πœƒ πœƒ  c o s s i n s i n c o s
  • B  πœƒ πœƒ βˆ’ πœƒ πœƒ  c o s s i n s i n c o s
  • C  πœƒ πœƒ πœƒ πœƒ  c o s s i n s i n c o s
  • D  πœƒ πœƒ πœƒ πœƒ  s i n c o s c o s s i n
  • E  πœƒ βˆ’ πœƒ πœƒ πœƒ  s i n c o s c o s s i n

What is the matrix 𝐡 ?

  • A  1 0 0 βˆ’ 1 
  • B  0 βˆ’ 1 βˆ’ 1 0 
  • C  βˆ’ 1 0 0 1 
  • D  0 1 1 0 
  • E  βˆ’ 1 0 0 βˆ’ 1 

What is the matrix 𝐴 𝐡 ?

  • A  πœƒ πœƒ πœƒ βˆ’ πœƒ  c o s s i n s i n c o s
  • B  πœƒ βˆ’ πœƒ πœƒ πœƒ  s i n c o s c o s s i n
  • C  πœƒ πœƒ βˆ’ πœƒ πœƒ  c o s s i n s i n c o s
  • D  πœƒ πœƒ πœƒ βˆ’ πœƒ  s i n c o s c o s s i n
  • E  πœƒ βˆ’ πœƒ πœƒ πœƒ  c o s s i n s i n c o s
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