Pure 1.7 - Linear transformations

Question 1

How would you define a transformation in two dimensions?

Answer 1

By describing how a general point with position vector
(𝑥) is transformed
(𝑦)
The new point is called the image

Question 2

What is a linear transformation?

Answer 2

A transformation that only involves linear terms in 𝑥 and 𝑦
Linear transformation always map the origin onto itself
Any linear transformation can be represented by a matrix
The linear transformation 𝑻:
(𝑥)→(𝑎𝑥 + 𝑏𝑦)
(𝑦)_(𝑐𝑥 + 𝑑𝑦)
can be represented by the matrix 𝐌=
(𝑎 𝑏)
(𝑐 𝑑)

Question 3

What are invariant points and lines?

Answer 3

An invariant point is a point which is mapped onto itself under a given transformation.
Invariant lines are lines which are mapped onto themselves under a given transformation.

Question 4

Describe the transformation represented by the matrix
(-1 0)
(0 1)

Answer 4

A reflection in the 𝑦-axis, points on the 𝑦-axis are invariant points, the lines 𝑥=0 and 𝑦=𝑘 are invariant lines.

Question 5

Describe the transformation represented by the matrix
(1 0)
(0 -1)

Answer 5

A reflection in the 𝑥-axis, points on the 𝑥-axis are invariant points, the lines 𝑦=0 and 𝑥=𝑘 are invariant lines.

Question 6

Describe the transformation represented by the matrix
(0 1)
(1 0)

Answer 6

A reflection in the line 𝑦=𝑥 points on the line 𝑦=𝑥 are invariant points and the lines 𝑦=𝑥 and 𝑦=-𝑥+𝑘 for any value of 𝑘 are invariant.

Question 7

Describe the transformation represented by the matrix
(cos(θ) -sin(θ))
(sin(θ) cos(θ))

Answer 7

A rotation through the angle θ anticlockwise about the origin. 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 8

Describe transformations represented by matrices of the type
(𝑎 0)
(0 𝑏)

Answer 8

It is a stretch of scale factor 𝑎 parallel to the 𝑥-axis and a stretch of scale factor 𝑏 parallel to the 𝑦-axis.
In the case that 𝑎=𝑏, the transformation is an enlargement with scale factor 𝑎.
For any stretch of this form, the 𝑥-axis and 𝑦-axis are invariant lines and the origin is an invariant point.
For a stretch parallel to the 𝑥-axis only, points on the 𝑦-axis are invariant points, and any line parallel to the 𝑥-axis is an invariant line.
For a stretch parallel to the 𝑦-axis only, points on the 𝑥-axis are invariant points, and any line parallel to the 𝑦-axis is an invariant line.

Question 9

For a linear transformation represented by the matrix 𝐌, what does det(𝐌) represent?

Answer 9

The scale factor for the change in area, sometimes called the area scale factor.

Question 10

How would you represent successive transformations, represent by the matrices 𝐏 and 𝐐 respectively.

Answer 10

The matrix 𝐐𝐏 represents the transformation 𝑄 followed by the transformation 𝑃.

Question 11

Describe the transformation represented by the matrix
(-1 0 0)
(0 1 0)
(0 0 1)

Answer 11

A reflection in the plane 𝑥=0.

Question 12

Describe the transformation represented by the matrix
( 1 0 0)
(0 -1 0)
(0 0 1)

Answer 12

A reflection in the plane 𝑦=0.

Question 13

Describe the transformation represented by the matrix
(1 0 0)
(0 1 0)
(0 0 -1)

Answer 13

A reflection in the plane 𝑧=0.

Question 14

Describe the transformation represented by the matrix
(1 __0 ____0__)
(0 cos(θ) -sin(θ) )
(0_sin(θ)_cos(θ))

Answer 14

A rotation, angle θ, about the 𝑥-axis.

Question 15

Describe the transformation represented by the matrix
(cos(θ) 0 sin(θ) )
(_ 0_ _ 1 ___0_ )
(-sin(θ)_0 cos(θ) )

Answer 15

A rotation, angle θ, about the 𝑦-axis.

Question 16

Describe the transformation represented by the matrix
(cos(θ) -sin(θ) 0)
( sin(θ) cos(θ) 0)
(__0___ 0___1)

Answer 16

A rotation, angle θ, about the 𝑧-axis.

Question 17

What is the effect of a transformation represented by the matrix 𝐀⁻¹.

Answer 17

It has the opposite effect of the transformation represented by the matrix 𝐀.