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 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
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- D
- E

What is the image of the point under this reflection?

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- 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
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- 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 .

**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 .