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Worksheet: Properties of Linear Transformations

Q1:

Consider the matrix where .

Find .

  • A
  • B
  • C
  • D
  • E

Find .

  • A
  • B2
  • C
  • D1
  • E0

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

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

Q2:

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?

  • Aan arrowhead with vertices ( 0 , 0 ) , ( 0 , 3 ) , ( 3 , 0 ) , and ( 3 , 3 )
  • Ba square with vertices ( 0 , 0 ) , ( 0 , 3 ) , ( 3 , 0 ) , and ( 3 , 3 )
  • Can arrowhead with vertices ( 0 , 0 ) , ( 0 , 3 ) , ( 3 , 0 ) , and ( 3 , 3 )
  • Da square with vertices ( 0 , 0 ) , ( 0 , 3 ) , ( 3 , 0 ) , and ( 3 , 3 )
  • Ea kite 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 center the origin
  • Ba stretch in the 𝑦 -direction
  • Ca stretch in the 𝑥 -direction
  • Da dilation with scale factor 3 and center the origin
  • Ea rotation about the origin by an angle of 3

Q3:

The vertex matrix of a square of side 1 shown is

Determine the vertex matrix of the image after a transformation by the matrix 1 2 2 1 , and state what geometric figure it is.

  • A 7 2 1 1 2 9 2 5 2 5 2 7 2 3 2 1 2 , a square
  • B 5 2 7 2 1 1 2 9 2 3 2 1 2 5 2 1 1 2 , a rectangle
  • C 5 2 7 2 1 1 2 9 2 3 2 1 2 5 2 1 1 2 , a parallelogram
  • D 7 2 1 1 2 9 2 5 2 5 2 7 2 3 2 1 2 , a rhombus

Q4:

Let 𝐿 be the transformation produced by a nonzero matrix with a zero determinant. What is the image of a unit square under 𝐿 ?

  • Aanother square
  • Ba single point
  • Ca parallelogram
  • Da line segment containing the origin
  • Ea rhombus

Q5:

Let 𝐿 be a linear transformation of 2 into itself with the property that 𝐿 6 4 = 2 2 and 𝐿 2 1 1 2 = 3 3 .

Using the fact that 6 4 = 6 4 , find 𝐿 6 4 .

  • A 2 2
  • B 2 2
  • C 6 4
  • D 2 2
  • E 6 4

Using the fact that 1 2 8 = 2 6 4 , find 𝐿 1 2 8 .

  • A 4 4
  • B 2 8 4 4
  • C 2 2
  • D 4 4
  • E 1 2 6

Find a vector 𝑣 so that 𝐿 ( 𝑣 ) = 1 1 .

  • A 2 1
  • B 3 2
  • C 3 2
  • D 2 2
  • E 1 2

What is 𝐿 ( 4 𝑣 + 𝑤 ) , where 𝑣 = 6 4 and 𝑤 = 2 1 1 2 ?

  • A 5 1 1
  • B 1 5 8
  • C 1 1 2 9
  • D 4 2 0
  • E 1 0 1 4

By considering suitable linear combinations of 6 4 and 2 1 4 , find 𝐿 1 0 and 𝐿 1 0 .

  • A 1 2 , 1 2
  • B 1 2 , 3 5
  • C 3 5 , 1 2
  • D 1 3 , 2 5
  • E 3 5 , 3 5

Find the matrix 𝑀 which represents the linear transformation 𝐿 .

  • A 𝑀 = 1 3 2 5
  • B 𝑀 = 3 5 3 5
  • C 𝑀 = 3 5 1 2
  • D 𝑀 = 1 2 3 5
  • E 𝑀 = 1 2 1 2