Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.
Start Practicing

Worksheet: 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 and are matrices, with representing a counterclockwise rotation of about the origin and representing a reflection in the -axis. What does the matrix represent?

  • Aa reflection in the line through the origin at a inclination
  • Ba reflection in the line through the origin at a inclination
  • Ca reflection in the line through the origin at a inclination
  • Da reflection in the line through the origin at a inclination
  • Ea reflection in the line through the origin at a 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 and are matrices, with representing a counterclockwise rotation of about the origin and representing a reflection in the -axis. What does the matrix represent?

  • Aa reflection in the line through the origin at a inclination
  • Ba reflection in the line through the origin at a inclination
  • Ca reflection in the line through the origin at a inclination
  • Da reflection in the line through the origin at a inclination
  • Ea reflection in the line through the origin at a 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 eigenvalues are , with corresponding eigenvector , and 1, with corresponding eigenvectors and .
  • B. Its eigenvalues are , with corresponding eigenvector , and 1, with corresponding eigenvectors and .
  • C. Its eigenvalues are , with corresponding eigenvector , and 1, with corresponding eigenvectors and .
  • D. Its only eigenvalue is 1, with corresponding eigenvectors and .
  • E. Its only eigenvalue is 1, with corresponding eigenvectors and .