Worksheet: Matrix Reflection

In this worksheet, we will practice finding the matrix of the linear transformation of reflecting vectors through a given axis.

Q1:

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

Q2:

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

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

Q3:

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

  • 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:

Suppose 𝐴 and 𝐵 are 2 × 2 matrices, with 𝐴 representing a counterclockwise rotation of 3 0 about the origin and 𝐵 representing a reflection in the 𝑥 -axis. What does the matrix 𝐵 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 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 4 5 inclination

Q6:

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

Q7:

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

Q8:

Suppose 𝐴 and 𝐵 are 2 × 2 matrices, with 𝐴 representing a counterclockwise rotation of 3 0 about the origin and 𝐵 representing a reflection in the 𝑥 -axis. What does the matrix 𝐴 𝐵 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 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 4 5 inclination

Q9:

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

  • A 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 .
  • B 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 .
  • C 𝑇 = 1 0 0 0 1 0 0 0 1 . Its eigenvalues are 1 , with corresponding eigenvector 0 0 1 , and 1, with corresponding eigenvectors 0 1 1 and 1 0 0 .
  • D 𝑇 = 1 0 0 0 1 0 0 0 1 . Its eigenvalues are 1 , with corresponding eigenvector 0 0 1 , and 1, with corresponding eigenvectors 0 1 0 and 1 0 0 .
  • E 𝑇 = 1 0 0 0 1 0 0 0 1 . Its eigenvalues are 1 , with corresponding eigenvector 0 0 1 , and 1, with corresponding eigenvectors 0 1 0 and 1 0 0 .

Q10:

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

  • A 𝑇 = 1 1 1 2 5 . Its eigenvalues are 2 with corresponding eigenvector 1 1 and 2 with corresponding eigenvector 1 1 .
  • B 𝑇 = 0 4 2 1 . Its eigenvalues are 2 with corresponding eigenvector 2 1 and 4 with corresponding eigenvector 1 1 .
  • C 𝑇 = 1 0 0 1 . Its eigenvalues are 1 with corresponding eigenvector 0 1 and 1 with corresponding eigenvector 1 1 .
  • D 𝑇 = 1 0 0 1 . Its eigenvalues are 1 with corresponding eigenvector 0 1 and 1 with corresponding eigenvector 1 0 .
  • E 𝑇 = 0 4 2 1 . Its eigenvalues are 2 with corresponding eigenvector 2 3 and 4 with corresponding eigenvector 1 1 .

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