Worksheet: Dilation Transformation Matrix

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In this worksheet, we will practice finding the matrix representation of dilations as geometric linear transformations.

Q1:

Consider the transformation represented by the matrix 3 0 0 3 .

What is the image of the square with vertices ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) , and ( 1 , 1 ) under this transformation?

  • Aa kite with vertices ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) , and ( 3 , 3 )
  • Ba kite with vertices ( 0 , 0 ) , ( 1 , 3 ) , ( 3 , 1 ) , and ( 3 , 3 )
  • Ca square with vertices ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) , and ( 1 , 1 )
  • Da square with vertices ( 0 , 0 ) , ( 0 , 3 ) , ( 3 , 0 ) , and ( 3 , 3 )
  • Ea square with vertices ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) , and ( 3 , 3 )

What geometric transformation does this matrix represent?

  • Aa dilation with scale factor 3 and centre the origin
  • Ba rotation about the origin by an angle of 3 ∘
  • Ca stretch in the 𝑦 -direction
  • Da stretch in the 𝑥 -direction
  • Ea dilation by a factor of 3 with its centre at the point ( 1 , 1 )

Q2:

Describe the geometric effect of the transformation produced by the matrix 0 − 3 3 0 .

  • Aa dilation with centre the origin and scale factor 3 followed by a − 9 0 ∘ rotation about the origin
  • Ba dilation with centre the origin and scale factor 3 followed by a reflection in the line 𝑦 = 𝑥
  • Ca dilation with centre the origin and scale factor 3 followed by a reflection in the line 𝑦 = − 𝑥
  • Da dilation with centre the origin and scale factor 3 followed by a 9 0 ∘ rotation about the origin
  • Ea dilation with centre the origin and scale factor 3 followed by a 1 8 0 ∘ rotation about the origin

Q3:

Which of the following compositions of transformations is represented by the matrix 0 − 2 − 2 0 ?

  • Aa dilation with centre the origin and scale factor 2 followed by a reflection in the line 𝑦 = 𝑥
  • Ba dilation with centre the origin and scale factor − 2 followed by a reflection in the line 𝑦 = − 𝑥
  • C a dilation with centre the origin and scale factor 2 followed by a reflection in the line 𝑦 = − 𝑥
  • Da dilation with centre the origin and scale factor − 2 followed by a reflection in the line 𝑦 = 𝑥
  • Ea rotation by 1 8 0 ∘ about the origin followed by a reflection in the line 𝑦 = 𝑥

Q4:

A dilation with center the origin is composed with a rotation about the origin to form a new linear transformation. The transformation formed sends the vector to .

Find the matrix representation of the transformation formed.

  • A
  • B
  • C
  • D
  • E

Find the scale factor of the original dilation.

  • Ascale factor = 13
  • Bscale factor =
  • Cscale factor = 154
  • Dscale factor = 169
  • Escale factor = 13

Q5:

The unit square, with vertices , and , is transformed by a rotation and then a dilation. Its image under this combined transformation is , as shown in the diagram.

What are the coordinates of ?

  • A
  • B
  • C
  • D
  • E

What is the matrix of the combined transformation?

  • A
  • B
  • C
  • D
  • E

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