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Lesson: Matrix Reflection

In this lesson, we will learn how to find the matrix of the linear transformation of reflecting vectors through a given axis.

Worksheet • 10 Questions

Q1:

Which of the following are necessary and sufficient conditions on , , , and for the matrix to represent a reflection?

  • A , ,
  • B , ,
  • C , ,
  • D , ,
  • E , ,

Q2:

Let 𝑇 be the linear transformation that reflects all vectors in ℝ 2 in the π‘₯ -axis. Represent 𝑇 as a matrix and find its eigenvalues and eigenvectors.

  • A 𝑇 = ο€Ό 1 0 0 βˆ’ 1  . Its eigenvalues are βˆ’ 1 with corresponding eigenvector ο€Ό 0 1  and 1 with corresponding eigenvector ο€Ό 1 0  .
  • B 𝑇 = ο€Ό 0 4 2 βˆ’ 1  . Its eigenvalues are 2 with corresponding eigenvector ο€Ό 2 3  and 4 with corresponding eigenvector ο€Ό 1 1  .
  • C 𝑇 = ο€Ό 0 4 2 βˆ’ 1  . Its eigenvalues are 2 with corresponding eigenvector ο€Ό 2 1  and 4 with corresponding eigenvector ο€Ό 1 1  .
  • D 𝑇 = ο€Ό 1 0 0 βˆ’ 1  . Its eigenvalues are βˆ’ 1 with corresponding eigenvector ο€Ό 0 1  and 1 with corresponding eigenvector ο€Ό 1 βˆ’ 1  .
  • E 𝑇 = ο€Ό 1 βˆ’ 1 1 2 βˆ’ 5  . Its eigenvalues are βˆ’ 2 with corresponding eigenvector ο€Ό βˆ’ 1 1  and 2 with corresponding eigenvector ο€Ό βˆ’ 1 1  .

Q3:

A vector in is rotated counterclockwise about the origin through an angle of , and the result is reflected in the -axis. Find, with respect to the standard basis, the matrix of this combined transformation.

  • A
  • B
  • C
  • D
  • E

Q4:

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

  • A
  • B
  • C
  • D
  • E

Q5:

Consider the linear transformation which maps a point to its reflection in the -axis.

Find the matrix which represents this transformation.

  • A
  • B
  • C
  • D
  • E

Where does this transformation map the point ?

  • A
  • B
  • C
  • D
  • E

Q6:

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

Q7:

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

  • A
  • B
  • C
  • D
  • E

Q8:

Consider the reflection in the line .

Find the matrix that represents this transformation.

  • A
  • B
  • C
  • D
  • E

What is the image of the point under this reflection?

  • A
  • B
  • C
  • D
  • E

Q9:

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

Q10:

Let 𝑇 be the linear transformation that reflects all vectors in ℝ 3 in the π‘₯ 𝑦 -plane. Represent 𝑇 as a matrix and find its eigenvalues and eigenvectors.

  • A 𝑇 =  1 0 0 0 1 0 0 0 βˆ’ 1  . Its eigenvalues are βˆ’ 1 , with corresponding eigenvector  0 0 1  , and 1, with corresponding eigenvectors  0 1 0  and  1 0 0  .
  • B 𝑇 =  1 0 0 0 βˆ’ 1 0 0 0 βˆ’ 1  . Its eigenvalues are βˆ’ 1 , with corresponding eigenvector  0 0 1  , and 1, with corresponding eigenvectors  0 1 0  and  1 0 0  .
  • C 𝑇 =  1 0 0 0 1 0 0 0 1  . Its only eigenvalue is 1, with corresponding eigenvectors  0 1 0  and  1 0 0  .
  • D 𝑇 =  1 0 0 0 1 0 0 0 βˆ’ 1  . Its eigenvalues are βˆ’ 1 , with corresponding eigenvector  0 0 1  , and 1, with corresponding eigenvectors  0 1 1  and  1 0 0  .
  • E 𝑇 =  1 0 0 0 1 0 0 0 1  . Its only eigenvalue is 1, with corresponding eigenvectors  0 1 0  and  1 1 1  .
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