Year 1 Chapter 7: Linear Transformations

Question 1

What does the following matrix represent?
(-1 0 )
( 0 1 )

Answer 1

A reflection in the y-axis.

Question 2

What does the following matrix represent?
( 1 0 )
( 0 -1 )

Answer 2

A reflection in the x-axis.

Question 3

What does the following matrix represent?
( 0 1 )
( 1 0 )

Answer 3

A reflection in the line y = x.

Question 4

What does the following matrix represent?
( 0 -1 )
(-1 0 )

Answer 4

A reflection in the line y = -x.

Question 5

What matrix represents a reflection in the y-axis?

Answer 5

(-1 0 )
( 0 1 )

Question 6

What matrix represents a reflection in the x-axis?

Answer 6

( 1 0 )
( 0 -1 )

Question 7

What matrix represents a reflection in the line y = x?

Answer 7

( 0 1 )
( 1 0 )

Question 8

What matrix represents a reflection in the line y = -x?

Answer 8

( 0 -1 )
(-1 0 )

Question 9

What does the following matrix represent?
( cosθ -sinθ )
( sinθ cosθ )

Answer 9

A rotation about the origin through θ anticlockwise.

Question 10

What matrix represents a rotation about the origin through θ anticlockwise?

Answer 10

( cosθ -sinθ )
( sinθ cosθ )

Question 11

For a rotation about the origin through θ anticlockwise, what is invariant?

Answer 11

The only invariant point is the origin (0, 0). For θ ≠ 180°, there are no invariant lines. For θ = 180°, any line passing through the origin is an invariant line.

Question 12

What does the following matrix represent?
( a 0 )
( 0 b )

Answer 12

A stretch of scale factor a parallel to the x-axis and a stretch of scale factor b parallel to the y-axis.

Question 13

What matrix represents a stretch of scale factor a parallel to the x-axis and a stretch of scale factor b parallel to the y-axis?

Answer 13

( a 0 )
( 0 b )

Question 14

For a stretch in the x-axis of a, and y-axis of b, what is invariant?

Answer 14

The x-axes and y-axes are invariant lines and the origin is an invariant point.

Question 15

For a stretch parallel to the x-axis, what is invariant?

Answer 15

Points on the y-axis are invariant points, and any line parallel to the x-axis is an invariant line.

Question 16

For a stretch parallel to the y-axis, what is invariant?

Answer 16

points on the x-axis are invariant points, and any line parallel to the y-axis is an invariant line.

Question 17

What does the determinant of a matrix represent?

Answer 17

The scale factor for the change in area.

Question 18

What does the matrix PQ represent?

Answer 18

A transformation of the matrix Q followed by a transformation of the matrix P.

Question 19

What does the following matrix represent?
(-1 0 0 )
( 0 1 0 )
( 0 0 1 )

Answer 19

A reflection in the plane x = 0.

Question 20

What does the following matrix represent?
( 1 0 0 )
( 0 -1 0 )
( 0 0 1 )

Answer 20

A reflection in the plane y = 0.

Question 21

What does the following matrix represent?
( 1 0 0 )
( 0 1 0 )
( 0 0 -1 )

Answer 21

A reflection in the plane z = 0.

Question 22

What matrix represents a reflection in the plane x = 0?

Answer 22

(-1 0 0 )
( 0 1 0 )
( 0 0 1 )

Question 23

What matrix represents a reflection in the plane y = 0?

Answer 23

( 1 0 0 )
( 0 -1 0 )
( 0 0 1 )

Question 24

What matrix represents a reflection in the plane z = 0?

Answer 24

( 1 0 0 )
( 0 1 0 )
( 0 0 -1 )

Question 25

What does the following matrix represent?
( 1 0 0 )
( 0 cosθ -sinθ )
( 0 sinθ cosθ )

Answer 25

A rotation about the x-axis through θ anticlockwise.

Question 26

What does the following matrix represent?
( cosθ 0 sinθ )
( 0 1 0 )
(-sinθ 0 cosθ )

Answer 26

A rotation about the y-axis through θ anticlockwise.

Question 27

What does the following matrix represent?
( cosθ -sinθ 0 )
( sinθ cosθ 0 )
( 0 0 1 )

Answer 27

A rotation about the z-axis through θ anticlockwise.

Question 28

What matrix represents a rotation about the x-axis through θ anticlockwise?

Answer 28

( 1 0 0 )
( 0 cosθ -sinθ )
( 0 sinθ cosθ )

Question 29

What matrix represents a rotation about the y-axis through θ anticlockwise?

Answer 29

( cosθ 0 sinθ )
( 0 1 0 )
(-sinθ 0 cosθ )

Question 30

What matrix represents a rotation about the z-axis through θ anticlockwise?

Answer 30

( cosθ -sinθ 0 )
( sinθ cosθ 0 )
( 0 0 1 )