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A Level Further Maths > Linear Transformations > Flashcards

Linear Transformations Flashcards

1
Q

What is the new point after a transformation called?

A

The image

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2
Q

What does a linear transformation mean?

In equations?

A

All variables in the matrix is in the form ax+by

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3
Q

What is a common mistake in identifying a linear transformation?

A

X+3 is not a linear transformation

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4
Q

What does a linear transformation always do?

A

It always maps the origin onto itself

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5
Q

What can all linear transformations be represented as?

A

A matrix

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6
Q

What is the general form to explain matrices in linear transformations?

What makes this statement true?

A
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7
Q

What is the notation for a transformed point?

A

Point S’ has been transformed from point S

Or: look at the image

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8
Q

How can all linear transformations be described?

A

They can be described on the effect that they have on the unit vectors:

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9
Q

What is the most important thing with 2x2 matrix linear transformations?

A
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10
Q

What does this infer: T(a+b)

A

Since T(a+b)=Ta+Tb you can add 2 image vectors to find the 4th vertex after a transformation has been done

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11
Q

When won’t change in a linear transformation?

A

The origin

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12
Q

What is an invariant point?

A

points that are mapped onto themselves under the given transformation

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13
Q

What is an invariant line?

A

lines that each point is mapped back onto the line
They can move along the line so be careful

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14
Q

What is always an invariant point?

A

The origin

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15
Q

What matrix represents a reflection in the y axis?
(2x2)

A
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16
Q

When doing a reflection in they y axis, where are the invariant points?

A

All points on the y axis

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17
Q

When doing a reflection in they y axis, which lines are invariant lines?

A

X=0 and y=k (for any value of k)

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18
Q

What matrix represents a reflection in the x axis?

A
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19
Q

When doing a reflection in the x axis, which points are invariant points?

A

All points on the x axis

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20
Q

When doing a reflection in the x axis, which lines are invariant lines?

A

The lines: y=0 and x=k (for any value of k)

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21
Q

State the matrices used for a reflection in the x and y axis

A
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22
Q

What 4 lines do you need to know the matrix for a reflection in?

A

y and x axis
y=x
y=-x

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23
Q

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

A
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24
Q

When doing a reflection in the line y=x, what are the invariant points?

A

Points on the line y=x

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25
Q

When doing a reflection in the line y=x, what are the invariant lines?

A

The lines y=x and y=-x+k (for any value of k) are invariant lines

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26
Q

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

A
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27
Q

When doing a reflection in the line y=-x, what are the invariant points?

A

Points on the line y=-x are invariant points

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28
Q

When doing a reflection in the line y=-x, what are the invariant lines?

A

The lines y=-x and y=x+k (for any value of k) are invariant lines

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29
Q

How do you show that some line (eg y=3x) is invariant under this transformation?

A
30
Q

What is the matrix which represents an anticlockwise rotation through angle , ø, around the origin

A
31
Q

What are the 4 reflection matrices that you need to know?
State the matrices

A
32
Q

What is extremely important to remember when using the rotation matrix?

A

It is in an anticlockwise direction

33
Q

When using the rotation matrix, what is the invariant point?

A

0,0 (the origin)

34
Q

When using the rotation matrix, when is there no invariant lines?

A

When ø≠180

35
Q

When using the rotation matrix, when are there invariant lines and what are they?

A

When ø=180
Any line passing through the origin is an invariant line when ø=180

36
Q

When has an enlargement occurred?

A

When both sides have been enlarged by the same factor

37
Q

What do you do if you are asked to describe a stretch?

A

You need to state the scale factor for both sides

38
Q

What is the stretch matrix?

A
39
Q

Explain the core concept of the stretch matrix?

A

There is a stretch of scale factor a parallel to the x axis
There is a stretch of scale factor b parallel to they y axis

40
Q

When is the stretch matrix going to result in an enlargement?

A

When a=b

41
Q

What are the invariant lines in the stretch matrix?

What is the invariant point?

A

The x and y axis are invariant lines

The origin is the invariant point

42
Q

What will the simpler matrix be for a stretch only parallel to the x axis?

A
43
Q

What will the simpler matrix be for a stretch only parallel to the y axis?

A
44
Q

For a stretch parallel to the x axis only, what are the invariant points and lines?

A

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

45
Q

For a stretch parallel to the y axis only, what are the invariant points and lines?

A

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

46
Q

What is the geometry difference between Reflections and stretches?

A

Reflections preserve their shape and area
Rotations preserve their shape and area
Stretches don’t preserve their shape or area

47
Q

For a linear transformation represented by matM, what does detM represent?

A

Scale factor for the change in area
Aka, the area scale factor

48
Q

Using the determinant ow do you know that a reflection has occurred?

A

DetM will be negative so the area scale factor will be negative

49
Q

How can you calculate the area scale factor?

A

Finding detM
The area scale factor will be the linear scale actor squared

50
Q

What do you know about a non zero singular 2x2 matrix representing a linear transformation?

A

It will map any point in the plane onto a straight line through the origin

51
Q

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

A
52
Q

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

A
53
Q

What matrix represents a rotation of angle θ around the x axis?

A
54
Q

What matrix represents a rotation of angle θ around the y axis?

A
55
Q

What matrix represents a rotation of angle θ around the z axis?

A
56
Q

Is the rotation around the x,y and z axis in the clockwise pr anticlockwise direction?

A

Anticlockwise

57
Q

How do you calculate the volume of a tetrahedron?

A
58
Q

In all cases how is θ measured in a rotation matrix?

A

In all cases θ is the angle measured anticlockwise when facing towards the origin from the positive side of the axis

59
Q

What does the matrix PQ represent?

A

The Matrix PQ represents the transformation of Q (with matrix Q) followed by the transformation of P (with the matrix P)

(Think about reversing the compound transformation)

60
Q

What is the only time that you will need to recognise that a matrix will be squared?

A

If C is represented by matM, then C followed by C will be represented by M^2

61
Q

What are the 3 unit vectors when you are in 3D?

A
62
Q
A
63
Q

In 3D what does x=0 represent?

A

They planer of x (a 2D shape not a line)

64
Q

Have a general idea of what the different planes look like?

A
65
Q

What 2 things do you need to remember about matrices?

A

AB≠BA

66
Q

How can we use the inverse of a matrix to our advantage?

A

The bottom bit is the important bit

67
Q

What do you need to remember to do when describing a rotation?

A

Say AROUND THE ORIGIN

68
Q

.

A
69
Q

What is the matrix for a reflection in the y axis?

A

100
0-10
001

70
Q

What is a tip for reflections in 3D matrices?

A

The only change is in that reflections unit vector

71
Q

What is a top for 3D rotation matrices?

A

That rotations unit vector stays the same

A Level Further Maths (20 decks)

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