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Worksheet: Introduction to Linear Transformations

Q1:

A translation of 3 units right and 2 units down can be described by the vector  3 βˆ’ 2  .

Describe the translation from point 𝐴 to point 𝐡 using a vector.

  • A  βˆ’ 5 βˆ’ 3 
  • B  3 5 
  • C
  • D  5 3 
  • E  3 2 

The point ( 1 , βˆ’ 4 ) is translated using the vector  βˆ’ 5 2  . What are the coordinates of its image?

  • A ( βˆ’ 5 , 2 )
  • B ( 6 , βˆ’ 6 )
  • C ( βˆ’ 2 , βˆ’ 4 )
  • D ( βˆ’ 4 , βˆ’ 2 )

Q2:

Consider the given figure.

The points 𝑂 ( 0 , 0 ) , 𝐴 ( 1 , 0 ) , 𝐡 ( 1 , 1 ) , and 𝐢 ( 0 , 1 ) are corners of the unit square. This square is reflected in the line 𝑂 𝐷 with equation 𝑦 = 1 2 π‘₯ to form the image 𝑂 𝐴 𝐡 𝐢 βˆ— βˆ— βˆ— .

As 𝐴 βˆ— is the image of 𝐴 in the line through 𝑂 and 𝐷 , π‘š ∠ 𝐴 𝑂 𝐴 = 2 π‘š ∠ 𝐷 𝑂 𝐴 βˆ— . Use this fact and the identity t a n t a n t a n 2 πœƒ = 2 πœƒ 1 βˆ’ πœƒ  to find the gradient and hence equation of βƒ–        βƒ— 𝑂 𝐴 βˆ— from the gradient of βƒ–      βƒ— 𝑂 𝐷 .

  • A 𝑦 = 2 3 π‘₯
  • B 𝑦 = βˆ’ 4 3 π‘₯
  • C 𝑦 = βˆ’ 2 3 π‘₯
  • D 𝑦 = 4 3 π‘₯
  • E 𝑦 = 3 4 π‘₯

Using the fact that βƒ–        βƒ— 𝑂 𝐢 βˆ— is perpendicular to βƒ–       βƒ— 𝑂 𝐴 βˆ— , find the equation of βƒ–        βƒ— 𝑂 𝐢 βˆ— .

  • A 𝑦 = βˆ’ 3 4 π‘₯
  • B 𝑦 = βˆ’ 4 3 π‘₯
  • C 𝑦 = 4 3 π‘₯
  • D 𝑦 = 3 4 π‘₯
  • E 𝑦 = βˆ’ 3 2 π‘₯

Using the fact that 𝑂 𝐢 = 𝑂 𝐴 = 1 βˆ— βˆ— , find the coordinates of 𝐢 βˆ— and 𝐴 βˆ— .

  • A 𝐢 = ο€Ό 1 6 2 5 , βˆ’ 9 2 5  βˆ— , 𝐴 = ο€Ό 9 2 5 , 1 6 2 5  βˆ—
  • B 𝐢 = ο€Ό βˆ’ 3 5 , βˆ’ 4 5  βˆ— , 𝐴 = ο€Ό 4 5 , 3 5  βˆ—
  • C 𝐢 = ο€Ό 4 5 , βˆ’ 3 5  βˆ— , 𝐴 = ο€Ό 3 5 , 4 5  βˆ—
  • D 𝐢 = ο€Ό 4 7 , βˆ’ 3 7  βˆ— , 𝐴 = ο€Ό 3 7 , 4 7  βˆ—
  • E 𝐢 = ο€Ό βˆ’ 3 7 , 4 7  βˆ— , 𝐴 = ο€Ό 4 7 , 3 7  βˆ—

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 = 1 2 π‘₯ .

  • A ⎑ ⎒ ⎒ ⎣ 3 5 4 5 4 5 βˆ’ 3 5 ⎀ βŽ₯ βŽ₯ ⎦
  • B ⎑ ⎒ ⎒ ⎣ 9 2 5 1 6 2 5 1 6 2 5 βˆ’ 9 2 5 ⎀ βŽ₯ βŽ₯ ⎦
  • C ⎑ ⎒ ⎒ ⎣ 4 7 3 7 3 7 βˆ’ 4 7 ⎀ βŽ₯ βŽ₯ ⎦
  • D ⎑ ⎒ ⎒ ⎣ 4 5 3 5 3 5 βˆ’ 4 5 ⎀ βŽ₯ βŽ₯ ⎦
  • E ⎑ ⎒ ⎒ ⎣ 3 7 4 7 4 7 βˆ’ 3 7 ⎀ βŽ₯ βŽ₯ ⎦

Q3:

A linear transformation of a plane sends the vector  1 0  to  𝑝 π‘ž  . If the transformation is a rotation, where does it send  0 1  ?

  • A  π‘ž 𝑝 
  • B  βˆ’ π‘ž 𝑝 
  • C  𝑝 π‘ž 
  • D  π‘ž βˆ’ 𝑝 
  • E  βˆ’ π‘ž βˆ’ 𝑝 

Q4:

Let 𝑇 ∈ 𝐿 ( 𝑉 ) be a linear transformation. Suppose the matrix for 𝑇 relative to a basis 𝐡 for 𝑉 is 𝑀 . Suppose 𝑃 is the transition matrix from another basis 𝐢 to 𝐡 . Determine the matrix for 𝑇 with respect to 𝐢 .

  • A 𝑃 𝑀
  • B 𝑃 𝑀 𝑃 βˆ’ 1
  • C 𝑀 𝑃
  • D 𝑃 𝑀 𝑃 βˆ’ 1

Q5:

Shape A has been translated to Shape B and then to Shape C.

Write a vector to represent the translation from Shape A to Shape B.

  • A  5 2 
  • B  βˆ’ 2 βˆ’ 5 
  • C  βˆ’ 2 βˆ’ 6 
  • D  βˆ’ 5 βˆ’ 2 
  • E  2 6 

Write a vector to represent the translation from Shape B to Shape C.

  • A  3 βˆ’ 4 
  • B  βˆ’ 2 βˆ’ 6 
  • C  βˆ’ 3 4 
  • D  βˆ’ 4 3 
  • E  2 6 

Write a vector to represent the translation from Shape C to Shape A.

  • A  3 βˆ’ 4 
  • B  6 2 
  • C  2 6 
  • D  βˆ’ 2 βˆ’ 6 
  • E  βˆ’ 3 4 