Pure 7: Linear Transformations

Question 1

What effect does the inverse of a linear transformation have?

Answer 1

It reverses the initial transformation

Question 2

What is always true for linear transformations?

Answer 2

The origin is mapped onto itself

Question 3

What is the formula for a rotation of θ anti-clockwise about the origin (2x2)?

Answer 3

cosθ -sinθ

sinθ cosθ

Question 4

How is a stretch represented (2x2)?

Answer 4

a 0
0 b
(Stretch scale factor a, parallel to x-axis and scale factor b, parallel to y-axis)

Question 5

When does a stretch represent an enlargement?

Answer 5

a 0
0 b
Where a=b

Question 6

What does the determinant of a 2x2 matrix represent?

Answer 6

The area scale factor

Question 7

How is a reflection in the plane x=0 represented?

Answer 7

-1 0 0
0 1 0
0 0 1

Question 8

How is a reflection in the plane y=0 represented?

Answer 8

1 0 0
0 -1 0
0 0 1

Question 9

How is a reflection in the plane z=0 represented?

Answer 9

1 0 0
0 1 0
0 0 -1

Question 10

How is a rotation θ anti-clockwise about the x-axis represented?

Answer 10

1 0 0
0 cosθ -sinθ
0 sinθ cosθ

Question 11

How is a rotation θ anticlockwise about the y-axis represented?

Answer 11

cosθ 0 sinθ
0 1 0
-sinθ 0 cosθ

Question 12

How is a rotation θ anti-clockwise about the z-axis represented?

Answer 12

cosθ -sinθ 0
sinθ cosθ 0
0 0 1

Question 13

How are successive transformations denoted?

Answer 13

PQ represents the transformation Q followed by the transformation P

Question 14

What are invariant lines/points?

Answer 14

Those that do not change under a transformation