# 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 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 .