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Worksheet: Finding the Matrix of the Linear Transformation of Reflecting Vectors through a Given Axis

Q1:

A vector in ℝ 2 is rotated counterclockwise about the origin through an angle of 2 πœ‹ 3 , and the result is reflected in the π‘₯ -axis. Find, with respect to the standard basis, the matrix of this combined transformation.

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

Q2:

Which of the following are necessary and sufficient conditions on π‘Ž , 𝑏 , 𝑐 , and 𝑑 for the matrix  π‘Ž 𝑏 𝑐 𝑑  to represent a reflection?

  • A 𝑏 = 𝑐 , 𝑑 = βˆ’ π‘Ž , π‘Ž βˆ’ 𝑐 = 1 2 2
  • B 𝑏 = βˆ’ 𝑐 , 𝑑 = βˆ’ π‘Ž , π‘Ž + 𝑐 = 1 2 2
  • C 𝑏 = 𝑐 , 𝑑 = π‘Ž , π‘Ž + 𝑐 = 1 2 2
  • D 𝑏 = 𝑐 , 𝑑 = βˆ’ π‘Ž , π‘Ž + 𝑐 = 1 2 2
  • E 𝑏 = βˆ’ 𝑐 , 𝑑 = βˆ’ π‘Ž , π‘Ž βˆ’ 𝑐 = 1 2 2

Q3:

A reflection in a line through the origin sends the vector  3 4  to  4 3  . Find the matrix representation of this reflection.

  • A ⎑ ⎒ ⎒ ⎣ 4 5 3 5 βˆ’ 3 5 4 5 ⎀ βŽ₯ βŽ₯ ⎦
  • B ⎑ ⎒ ⎒ ⎣ 2 4 2 5 7 2 5 βˆ’ 7 2 5 2 4 2 5 ⎀ βŽ₯ βŽ₯ ⎦
  • C ⎑ ⎒ ⎒ ⎣ 2 4 2 5 7 2 5 7 2 5 βˆ’ 2 4 2 5 ⎀ βŽ₯ βŽ₯ ⎦
  • D  0 1 1 0 
  • E  1 1 0 1 

Q4:

A reflection in a line through the origin sends the vector  3 4  to  4 βˆ’ 3  . Find the matrix representation of this reflection.

  • A ⎑ ⎒ ⎒ ⎣ 4 5 3 5 βˆ’ 3 5 4 5 ⎀ βŽ₯ βŽ₯ ⎦
  • B ⎑ ⎒ ⎒ ⎣ 2 4 2 5 7 2 5 βˆ’ 7 2 5 2 4 2 5 ⎀ βŽ₯ βŽ₯ ⎦
  • C  0 1 1 0 
  • D ⎑ ⎒ ⎒ ⎣ 2 4 2 5 7 2 5 7 2 5 βˆ’ 2 4 2 5 ⎀ βŽ₯ βŽ₯ ⎦
  • E ⎑ ⎒ ⎒ ⎣ 4 5 3 5 3 5 βˆ’ 4 5 ⎀ βŽ₯ βŽ₯ ⎦

Q5:

Suppose 𝐴 and 𝐡 are 2 Γ— 2 matrices, with 𝐴 representing a counterclockwise rotation of 3 0 ∘ about the origin and 𝐡 representing a reflection in the π‘₯ -axis. What does the matrix 𝐡 𝐴 represent?

  • Aa reflection in the line through the origin at a 7 5 ∘ inclination
  • Ba reflection in the line through the origin at a 1 5 ∘ inclination
  • Ca reflection in the line through the origin at a βˆ’ 7 5 ∘ inclination
  • Da reflection in the line through the origin at a βˆ’ 1 5 ∘ inclination
  • Ea reflection in the line through the origin at a βˆ’ 4 5 ∘ inclination

Q6:

Consider the reflection in the line 𝑦 = 1 2 π‘₯ .

Find the matrix that represents this transformation.

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

What is the image of the point ( 1 2 , 5 ) under this reflection?

  • A ο€Ό 5 6 5 , 3 3 5 
  • B ο€Ό 1 7 2 , 7 2 
  • C ο€Ό 5 6 5 , 6 3 5 
  • D ο€Ό 6 3 5 , 1 6 5 
  • E ο€Ό 7 2 , 1 7 2 

Q7:

Consider the linear transformation which maps a point to its reflection in the π‘₯ -axis.

Find the matrix 𝐴 which represents this transformation.

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

Where does this transformation map the point ( 2 , βˆ’ 3 ) ?

  • A ( 2 , 3 )
  • B ( βˆ’ 2 , 3 )
  • C ( βˆ’ 2 , βˆ’ 3 )
  • D ( βˆ’ 2 , 3 )
  • E ( 2 , βˆ’ 3 )

Q8:

Suppose 𝐴 and 𝐡 are 2 Γ— 2 matrices, with 𝐴 representing a counterclockwise rotation of 3 0 ∘ about the origin and 𝐡 representing a reflection in the π‘₯ -axis. What does the matrix 𝐴 𝐡 represent?

  • Aa reflection in the line through the origin at a 7 5 ∘ inclination
  • Ba reflection in the line through the origin at a βˆ’ 1 5 ∘ inclination
  • Ca reflection in the line through the origin at a βˆ’ 7 5 ∘ inclination
  • Da reflection in the line through the origin at a 1 5 ∘ inclination
  • Ea reflection in the line through the origin at a βˆ’ 4 5 ∘ inclination

Q9:

Let be the linear transformation that reflects all vectors in in the -plane. Represent as a matrix and find its eigenvalues and eigenvectors.

  • A . Its only eigenvalue is 1, with corresponding eigenvectors and .
  • B . Its only eigenvalue is 1, with corresponding eigenvectors and .
  • C . Its eigenvalues are , with corresponding eigenvector , and 1, with corresponding eigenvectors and .
  • D . Its eigenvalues are , with corresponding eigenvector , and 1, with corresponding eigenvectors and .
  • E . Its eigenvalues are , with corresponding eigenvector , and 1, with corresponding eigenvectors and .

Q10:

Let be the linear transformation that reflects all vectors in in the -axis. Represent as a matrix and find its eigenvalues and eigenvectors.

  • A . Its eigenvalues are with corresponding eigenvector and 2 with corresponding eigenvector .
  • B . Its eigenvalues are 2 with corresponding eigenvector and 4 with corresponding eigenvector .
  • C . Its eigenvalues are with corresponding eigenvector and 1 with corresponding eigenvector .
  • D . Its eigenvalues are with corresponding eigenvector and 1 with corresponding eigenvector .
  • E . Its eigenvalues are 2 with corresponding eigenvector and 4 with corresponding eigenvector .