# Worksheet: Linear Transformations in Planes: Reflection

In this worksheet, we will practice finding the matrix of linear transformation of reflection along the x- or y-axis or the line of a given equation and the image of a vector under the reflection.

**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: **

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

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

**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: **

Consider the given figure.

The points , , , and are corners of the unit square. This square is reflected in the line with equation to form the image .

As is the image of in the line through and , . Use this fact and the identity to find the gradient and hence equation of from the gradient of .

- A
- B
- C
- D
- E

Using the fact that is perpendicular to , find the equation of .

- A
- B
- C
- D
- E

Using the fact that , find the coordinates of and .

- A ,
- B ,
- C ,
- D ,
- E ,

Using the fact that a reflection in a line through the origin is a linear transformation, find the matrix which represents reflection in the line .

- A
- B
- C
- D
- E

**Q7: **

Consider the matrix where .

Find .

- A
- B
- C
- D
- E

Find .

- A2
- B1
- C
- D0
- E

By drawing the image of the unit square under the transformation, identify the geometrical transformation this matrix corresponds to.

- Aa projection onto the line
- Ba rotation by clockwise about the point
- Ca reflection in the line
- Da rotation of clockwise about the origin
- Ea reflection in the line

**Q8: **

Consider the given figure.

The points , , , and are corners of the unit square. This square is reflected in the line with equation to form the image .

As is the image of in the line through and , . Use this fact and the identity to find the gradient and hence equation of from the gradient of .

- A
- B
- C
- D
- E

Using the fact that is perpendicular to , find the equation of .

- A
- B
- C
- D
- E

Using the fact that , find the coordinates of and .

- A ,
- B ,
- C ,
- D ,
- E ,

Using the fact that a reflection in a line through the origin is a linear transformation, find the matrix which represents reflection in the line .

- A
- B
- C
- D
- E