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 ๐‘Ž, ๐‘, ๐‘, and ๐‘‘ for the matrix ๏”๐‘Ž๐‘๐‘๐‘‘๏  to represent a reflection?

  • A๐‘=๐‘, ๐‘‘=โˆ’๐‘Ž, ๐‘Žโˆ’๐‘=1๏Šจ๏Šจ
  • B๐‘=โˆ’๐‘, ๐‘‘=โˆ’๐‘Ž, ๐‘Žโˆ’๐‘=1๏Šจ๏Šจ
  • C๐‘=โˆ’๐‘, ๐‘‘=โˆ’๐‘Ž, ๐‘Ž+๐‘=1๏Šจ๏Šจ
  • D๐‘=๐‘, ๐‘‘=๐‘Ž, ๐‘Ž+๐‘=1๏Šจ๏Šจ
  • E๐‘=๐‘, ๐‘‘=โˆ’๐‘Ž, ๐‘Ž+๐‘=1๏Šจ๏Šจ

Q2:

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

  • AโŽกโŽขโŽขโŽฃ2425725โˆ’7252425โŽคโŽฅโŽฅโŽฆ
  • B๏”1101๏ 
  • C๏”0110๏ 
  • DโŽกโŽขโŽขโŽฃ4535โˆ’3545โŽคโŽฅโŽฅโŽฆ
  • EโŽกโŽขโŽขโŽฃ2425725725โˆ’2425โŽคโŽฅโŽฅโŽฆ

Q3:

A reflection in a line through the origin sends the vector ๏”34๏  to ๏”4โˆ’3๏ . Find the matrix representation of this reflection.

  • AโŽกโŽขโŽขโŽฃ2425725โˆ’7252425โŽคโŽฅโŽฅโŽฆ
  • BโŽกโŽขโŽขโŽฃ453535โˆ’45โŽคโŽฅโŽฅโŽฆ
  • CโŽกโŽขโŽขโŽฃ4535โˆ’3545โŽคโŽฅโŽฅโŽฆ
  • D๏”0110๏ 
  • EโŽกโŽขโŽขโŽฃ2425725725โˆ’2425โŽคโŽฅโŽฅโŽฆ

Q4:

Consider the reflection in the line ๐‘ฆ=12๐‘ฅ.

Find the matrix that represents this transformation.

  • AโŽกโŽขโŽขโŽฃ35454535โŽคโŽฅโŽฅโŽฆ
  • BโŽกโŽขโŽขโŽฃ12โˆ’121212โŽคโŽฅโŽฅโŽฆ
  • CโŽกโŽขโŽขโŽฃ354545โˆ’35โŽคโŽฅโŽฅโŽฆ
  • DโŽกโŽขโŽขโŽฃ453535โˆ’45โŽคโŽฅโŽฅโŽฆ
  • EโŽกโŽขโŽขโŽฃ121212โˆ’12โŽคโŽฅโŽฅโŽฆ

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

  • A๏€ผ172,72๏ˆ
  • B๏€ผ565,335๏ˆ
  • C๏€ผ72,172๏ˆ
  • D๏€ผ565,635๏ˆ
  • E๏€ผ635,165๏ˆ

Q5:

Consider the linear transformation which maps a point to its reflection in the ๐‘ฅ-axis.

Find the matrix ๐ด which represents this transformation.

  • A๐ด=๏”10โˆ’11๏ 
  • B๐ด=๏”1001๏ 
  • C๐ด=๏”โˆ’101โˆ’1๏ 
  • D๐ด=๏”1101๏ 
  • E๐ด=๏”100โˆ’1๏ 

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

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

Q6:

Consider the given figure.

The points ๐‘‚(0,0), ๐ด(1,0), ๐ต(1,1), and ๐ถ(0,1) are corners of the unit square. This square is reflected in the line ๐‘‚๐ท with equation ๐‘ฆ=12๐‘ฅ to form the image ๐‘‚๐ด๐ต๐ถโˆ—โˆ—โˆ—.

As ๐ดโˆ— is the image of ๐ด in the line through ๐‘‚ and ๐ท, ๐‘šโˆ ๐ด๐‘‚๐ด=2๐‘šโˆ ๐ท๐‘‚๐ดโˆ—. Use this fact and the identity tantantan2๐œƒ=2๐œƒ1โˆ’๐œƒ๏Šจ to find the gradient and hence equation of โƒ–๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉโƒ—๐‘‚๐ดโˆ— from the gradient of โƒ–๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉโƒ—๐‘‚๐ท.

  • A๐‘ฆ=43๐‘ฅ
  • B๐‘ฆ=โˆ’23๐‘ฅ
  • C๐‘ฆ=34๐‘ฅ
  • D๐‘ฆ=โˆ’43๐‘ฅ
  • E๐‘ฆ=23๐‘ฅ

Using the fact that โƒ–๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉโƒ—๐‘‚๐ถโˆ— is perpendicular to โƒ–๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉโƒ—๐‘‚๐ดโˆ—, find the equation of โƒ–๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉโƒ—๐‘‚๐ถโˆ—.

  • A๐‘ฆ=โˆ’43๐‘ฅ
  • B๐‘ฆ=โˆ’32๐‘ฅ
  • C๐‘ฆ=43๐‘ฅ
  • D๐‘ฆ=34๐‘ฅ
  • E๐‘ฆ=โˆ’34๐‘ฅ

Using the fact that ๐‘‚๐ถ=๐‘‚๐ด=1โˆ—โˆ—, find the coordinates of ๐ถโˆ— and ๐ดโˆ—.

  • A๐ถ=๏€ผ1625,โˆ’925๏ˆโˆ—, ๐ด=๏€ผ925,1625๏ˆโˆ—
  • B๐ถ=๏€ผโˆ’35,โˆ’45๏ˆโˆ—, ๐ด=๏€ผ45,35๏ˆโˆ—
  • C๐ถ=๏€ผ45,โˆ’35๏ˆโˆ—, ๐ด=๏€ผ35,45๏ˆโˆ—
  • D๐ถ=๏€ผ47,โˆ’37๏ˆโˆ—, ๐ด=๏€ผ37,47๏ˆโˆ—
  • E๐ถ=๏€ผโˆ’37,47๏ˆโˆ—, ๐ด=๏€ผ47,37๏ˆโˆ—

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 ๐‘ฆ=12๐‘ฅ.

  • AโŽกโŽขโŽขโŽฃ453535โˆ’45โŽคโŽฅโŽฅโŽฆ
  • BโŽกโŽขโŽขโŽฃ473737โˆ’47โŽคโŽฅโŽฅโŽฆ
  • CโŽกโŽขโŽขโŽฃ92516251625โˆ’925โŽคโŽฅโŽฅโŽฆ
  • DโŽกโŽขโŽขโŽฃ354545โˆ’35โŽคโŽฅโŽฅโŽฆ
  • EโŽกโŽขโŽขโŽฃ374747โˆ’37โŽคโŽฅโŽฅโŽฆ

Q7:

Consider the matrix ๐‘€=๏“๐›ผ๐›ผ๐›ผโˆ’๐›ผ๏Ÿ where ๐›ผ=โˆš22.

Find ๐‘€๏Šจ.

  • A๏”100โˆ’1๏ 
  • B๏”1001๏ 
  • C๏”โˆ’1001๏ 
  • D๏”1111๏ 
  • EโŽกโŽขโŽขโŽฃ12121212โŽคโŽฅโŽฅโŽฆ

Find det(๐‘€).

  • A2
  • B1
  • C12
  • D0
  • Eโˆ’1

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

  • Aa projection onto the line ๐‘ฆ=๐‘ฅ
  • Ba rotation by 45โˆ˜ clockwise about the point (1,0)
  • Ca reflection in the line ๐‘ฆ=(22.5)๐‘ฅtanโˆ˜
  • Da rotation of 45โˆ˜ clockwise about the origin
  • Ea reflection in the line ๐‘ฆ=๐‘ฅ

Q8:

Consider the given figure.

The points ๐‘‚(0,0), ๐ด(1,0), ๐ต(1,1), and ๐ถ(0,1) are corners of the unit square. This square is reflected in the line ๐‘‚๐ท with equation ๐‘ฆ=๐‘˜๐‘ฅ to form the image ๐‘‚๐ด๐ต๐ถโˆ—โˆ—โˆ—.

As ๐ดโˆ— is the image of ๐ด in the line through ๐‘‚ and ๐ท, ๐‘šโˆ ๐ด๐‘‚๐ด=2๐‘šโˆ ๐ท๐‘‚๐ดโˆ—. Use this fact and the identity tantantan2๐œƒ=2๐œƒ1โˆ’๐œƒ๏Šจ to find the gradient and hence equation of โƒ–๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉโƒ—๐‘‚๐ดโˆ— from the gradient of โƒ–๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉโƒ—๐‘‚๐ท.

  • A๐‘ฆ=2๐‘˜1โˆ’๐‘˜๐‘ฅ๏Šจ
  • B๐‘ฆ=2๐‘˜1+๐‘˜๐‘ฅ๏Šจ
  • C๐‘ฆ=2๐‘˜๐‘˜โˆ’1๐‘ฅ๏Šจ
  • D๐‘ฆ=๐‘˜๐‘˜โˆ’1๐‘ฅ๏Šจ
  • E๐‘ฆ=๐‘˜1โˆ’๐‘˜๐‘ฅ๏Šจ

Using the fact that โƒ–๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉโƒ—๐‘‚๐ถโˆ— is perpendicular to โƒ–๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉโƒ—๐‘‚๐ดโˆ—, find the equation of โƒ–๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉ๏ƒฉโƒ—๐‘‚๐ถโˆ—.

  • A๐‘ฆ=๐‘˜โˆ’12๐‘˜๐‘ฅ๏Šจ
  • B๐‘ฆ=1โˆ’๐‘˜2๐‘˜๐‘ฅ๏Šจ
  • C๐‘ฆ=2๐‘˜1โˆ’๐‘˜๐‘ฅ๏Šจ
  • D๐‘ฆ=๐‘˜โˆ’12๐‘˜๐‘ฅ๏Šจ
  • E๐‘ฆ=2๐‘˜๐‘˜โˆ’1๐‘ฅ๏Šจ

Using the fact that ๐‘‚๐ถ=๐‘‚๐ด=1โˆ—โˆ—, find the coordinates of ๐ถโˆ— and ๐ดโˆ—.

  • A๐ถ=๏€พ2๐‘˜1+๐‘˜,๐‘˜โˆ’11+๐‘˜๏Šโˆ—๏Šจ, ๐ด=๏€พ1โˆ’๐‘˜1+๐‘˜,2๐‘˜1+๐‘˜๏Šโˆ—๏Šจ
  • B๐ถ=๏€พ๐‘˜โˆ’11+๐‘˜,2๐‘˜1+๐‘˜๏Šโˆ—๏Šจ๏Šจ๏Šจ, ๐ด=๏€พ2๐‘˜1+๐‘˜,1โˆ’๐‘˜1+๐‘˜๏Šโˆ—๏Šจ๏Šจ๏Šจ
  • C๐ถ=๏€พ๐‘˜1+๐‘˜,๐‘˜โˆ’11+๐‘˜๏Šโˆ—๏Šจ๏Šจ๏Šจ, ๐ด=๏€พ1โˆ’๐‘˜1+๐‘˜,๐‘˜1+๐‘˜๏Šโˆ—๏Šจ๏Šจ๏Šจ
  • D๐ถ=๏€พ2๐‘˜1+๐‘˜,๐‘˜โˆ’11+๐‘˜๏Šโˆ—๏Šจ๏Šจ๏Šจ, ๐ด=๏€พ1โˆ’๐‘˜1+๐‘˜,2๐‘˜1+๐‘˜๏Šโˆ—๏Šจ๏Šจ๏Šจ
  • E๐ถ=๏€พ๐‘˜โˆ’11+๐‘˜,2๐‘˜1+๐‘˜๏Šโˆ—๏Šจ, ๐ด=๏€พ2๐‘˜1+๐‘˜,1โˆ’๐‘˜1+๐‘˜๏Šโˆ—๏Šจ

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 ๐‘ฆ=๐‘˜๐‘ฅ.

  • AโŽกโŽขโŽขโŽขโŽฃ1โˆ’๐‘˜1+๐‘˜2๐‘˜1+๐‘˜2๐‘˜1+๐‘˜๐‘˜โˆ’11+๐‘˜โŽคโŽฅโŽฅโŽฅโŽฆ๏Šจ๏Šจ๏Šจ๏Šจ๏Šจ๏Šจ
  • BโŽกโŽขโŽขโŽขโŽฃ1โˆ’๐‘˜1+๐‘˜2๐‘˜1+๐‘˜2๐‘˜1+๐‘˜๐‘˜โˆ’11+๐‘˜โŽคโŽฅโŽฅโŽฅโŽฆ๏Šจ๏Šจ
  • CโŽกโŽขโŽขโŽขโŽฃ1โˆ’๐‘˜1+๐‘˜2๐‘˜1+๐‘˜2๐‘˜1+๐‘˜1โˆ’๐‘˜1+๐‘˜โŽคโŽฅโŽฅโŽฅโŽฆ๏Šจ๏Šจ๏Šจ๏Šจ๏Šจ๏Šจ
  • DโŽกโŽขโŽขโŽขโŽฃ1+๐‘˜1โˆ’๐‘˜1+๐‘˜2๐‘˜1+๐‘˜2๐‘˜1+๐‘˜๐‘˜โˆ’1โŽคโŽฅโŽฅโŽฅโŽฆ๏Šจ๏Šจ๏Šจ๏Šจ๏Šจ๏Šจ
  • EโŽกโŽขโŽขโŽขโŽฃ1โˆ’๐‘˜1+๐‘˜2๐‘˜1+๐‘˜โˆ’2๐‘˜1+๐‘˜1โˆ’๐‘˜1+๐‘˜โŽคโŽฅโŽฅโŽฅโŽฆ๏Šจ๏Šจ๏Šจ๏Šจ๏Šจ๏Šจ

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