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 𝑎, 𝑏, 𝑐, and 𝑑 for the matrix 𝑎𝑏𝑐𝑑 to represent a reflection?

  • A𝑏=𝑐, 𝑑=𝑎, 𝑎𝑐=1
  • B𝑏=𝑐, 𝑑=𝑎, 𝑎𝑐=1
  • C𝑏=𝑐, 𝑑=𝑎, 𝑎+𝑐=1
  • D𝑏=𝑐, 𝑑=𝑎, 𝑎+𝑐=1
  • E𝑏=𝑐, 𝑑=𝑎, 𝑎+𝑐=1

Q2:

A reflection in a line through the origin sends the vector 34 to 43. Find the matrix representation of this reflection.

  • A24257257252425
  • B1101
  • C0110
  • D45353545
  • E24257257252425

Q3:

A reflection in a line through the origin sends the vector 34 to 43. Find the matrix representation of this reflection.

  • A24257257252425
  • B45353545
  • C45353545
  • D0110
  • E24257257252425

Q4:

Consider the reflection in the line 𝑦=12𝑥.

Find the matrix that represents this transformation.

  • A35454535
  • B12121212
  • C35454535
  • D45353545
  • E12121212

What is the image of the point (12,5) under this reflection?

  • A172,72
  • B565,335
  • C72,172
  • D565,635
  • E635,165

Q5:

Consider the linear transformation which maps a point to its reflection in the 𝑥-axis.

Find the matrix 𝐴 which represents this transformation.

  • A𝐴=1011
  • B𝐴=1001
  • C𝐴=1011
  • D𝐴=1101
  • E𝐴=1001

Where does this transformation map the point (2,3)?

  • A(2,3)
  • B(2,3)
  • C(2,3)
  • D(2,3)
  • E(2,3)

Q6:

Consider the given figure.

The points 𝑂(0,0), 𝐴(1,0), 𝐵(1,1), and 𝐶(0,1) are corners of the unit square. This square is reflected in the line 𝑂𝐷 with equation 𝑦=12𝑥 to form the image 𝑂𝐴𝐵𝐶.

As 𝐴 is the image of 𝐴 in the line through 𝑂 and 𝐷, 𝑚𝐴𝑂𝐴=2𝑚𝐷𝑂𝐴. Use this fact and the identity tantantan2𝜃=2𝜃1𝜃 to find the gradient and hence equation of 𝑂𝐴 from the gradient of 𝑂𝐷.

  • A𝑦=43𝑥
  • B𝑦=23𝑥
  • C𝑦=34𝑥
  • D𝑦=43𝑥
  • E𝑦=23𝑥

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

  • A𝑦=43𝑥
  • B𝑦=32𝑥
  • C𝑦=43𝑥
  • D𝑦=34𝑥
  • E𝑦=34𝑥

Using the fact that 𝑂𝐶=𝑂𝐴=1, find the coordinates of 𝐶 and 𝐴.

  • A𝐶=1625,925, 𝐴=925,1625
  • B𝐶=35,45, 𝐴=45,35
  • C𝐶=45,35, 𝐴=35,45
  • D𝐶=47,37, 𝐴=37,47
  • E𝐶=37,47, 𝐴=47,37

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 𝑦=12𝑥.

  • A45353545
  • B47373747
  • C92516251625925
  • D35454535
  • E37474737

Q7:

Consider the matrix 𝑀=𝛼𝛼𝛼𝛼 where 𝛼=22.

Find 𝑀.

  • A1001
  • B1001
  • C1001
  • D1111
  • E12121212

Find det(𝑀).

  • A2
  • B1
  • C12
  • D0
  • E1

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

  • Aa projection onto the line 𝑦=𝑥
  • Ba rotation by 45 clockwise about the point (1,0)
  • Ca reflection in the line 𝑦=(22.5)𝑥tan
  • Da rotation of 45 clockwise about the origin
  • Ea reflection in the line 𝑦=𝑥

Q8:

Consider the given figure.

The points 𝑂(0,0), 𝐴(1,0), 𝐵(1,1), and 𝐶(0,1) are corners of the unit square. This square is reflected in the line 𝑂𝐷 with equation 𝑦=𝑘𝑥 to form the image 𝑂𝐴𝐵𝐶.

As 𝐴 is the image of 𝐴 in the line through 𝑂 and 𝐷, 𝑚𝐴𝑂𝐴=2𝑚𝐷𝑂𝐴. Use this fact and the identity tantantan2𝜃=2𝜃1𝜃 to find the gradient and hence equation of 𝑂𝐴 from the gradient of 𝑂𝐷.

  • A𝑦=2𝑘1𝑘𝑥
  • B𝑦=2𝑘1+𝑘𝑥
  • C𝑦=2𝑘𝑘1𝑥
  • D𝑦=𝑘𝑘1𝑥
  • E𝑦=𝑘1𝑘𝑥

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

  • A𝑦=𝑘12𝑘𝑥
  • B𝑦=1𝑘2𝑘𝑥
  • C𝑦=2𝑘1𝑘𝑥
  • D𝑦=𝑘12𝑘𝑥
  • E𝑦=2𝑘𝑘1𝑥

Using the fact that 𝑂𝐶=𝑂𝐴=1, find the coordinates of 𝐶 and 𝐴.

  • A𝐶=2𝑘1+𝑘,𝑘11+𝑘, 𝐴=1𝑘1+𝑘,2𝑘1+𝑘
  • B𝐶=𝑘11+𝑘,2𝑘1+𝑘, 𝐴=2𝑘1+𝑘,1𝑘1+𝑘
  • C𝐶=𝑘1+𝑘,𝑘11+𝑘, 𝐴=1𝑘1+𝑘,𝑘1+𝑘
  • D𝐶=2𝑘1+𝑘,𝑘11+𝑘, 𝐴=1𝑘1+𝑘,2𝑘1+𝑘
  • E𝐶=𝑘11+𝑘,2𝑘1+𝑘, 𝐴=2𝑘1+𝑘,1𝑘1+𝑘

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 𝑦=𝑘𝑥.

  • A1𝑘1+𝑘2𝑘1+𝑘2𝑘1+𝑘𝑘11+𝑘
  • B1𝑘1+𝑘2𝑘1+𝑘2𝑘1+𝑘𝑘11+𝑘
  • C1𝑘1+𝑘2𝑘1+𝑘2𝑘1+𝑘1𝑘1+𝑘
  • D1+𝑘1𝑘1+𝑘2𝑘1+𝑘2𝑘1+𝑘𝑘1
  • E1𝑘1+𝑘2𝑘1+𝑘2𝑘1+𝑘1𝑘1+𝑘

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