Transcript Matrix Transformations

IGCSE FM Matrix Transformations
Dr J Frost ([email protected])
www.drfrostmaths.com
The specification:
Last modified: 3rd January 2016
Introduction
A matrix (plural: matrices) is simply an ‘array’ of numbers, e.g.
On a simple level, a matrix is simply a way to organise values into rows and columns,
and represent these multiple values as a single structure.
For the purposes of IGCSE Further Maths, you should understand matrices as a
way to transform points.
Matrices are particularly useful in
3D graphics, as matrices can be
used to carry out
rotations/enlargements (useful for
changing the camera angle) or
project into a 2D ‘viewing’ plane.
(Just for Fun) Using matrices to represent data
This is a scene from the film Good Will?Hunting.
Maths professor Lambeau poses a “difficult”* problem for his graduate students from
algebraic graph theory, the first part asking for a matrix representation of this graph.
Matt Damon anonymously solves the problem while on a cleaning shift.
In an ‘adjacency matrix’, the
number in the ith row and jth
column is the number of
edges directly connecting
node (i.e. dot) i to dot j
?
* It really isn’t.
Using matrices to represent data
In my 4th year undergraduate dissertation, I used matrices to help ‘learn’ mark schemes from
GCSE biology scripts. Matrix algebra helped me to initially determine how words (and more
complex semantic information) tended to occur together with other words.
Matrix Algebra
Matrix Fundamentals
Understand the dimensions of a matrix, and operations on
matrices, such as addition, scalar multiplication and matrix
multiplication.
Matrix Fundamentals
#1 Dimensions of Matrices
The dimension of a matrix is its size, in terms of its number of rows and columns.
Matrix
Dimensions
23
31
?
13
?
Matrix Fundamentals
#2 Notation/Names for Matrices
A matrix can have square or curvy brackets*.
Matrix
Column Vector
Row Vector
(The vector you know
and love)
So a matrix with one column is simply a vector in the usual sense.
* The textbook only uses curvy.
Matrix Fundamentals
#3 Variables for Matrices
If we wish a variable to represent a matrix, we use bold, capital letters.
1
𝑨= 6
−3
𝑪 = 𝑷𝟐 𝑻𝑷
Matrix Fundamentals
#4 Adding/Subtracting Matrices
Simply add/subtract the corresponding elements of each matrix.
They must be of the same dimension.
?
?
Matrix Fundamentals
#5 Scalar Multiplication
A scalar is a number which can ‘scale’ the elements inside a matrix/vector.
1
2
3
?
?
?
Matrix Fundamentals
#6 Matrix Multiplication
This is where things get slightly more complicated...
Now repeat for the next row of the left matrix...
1
2
7
0 3 -2
8 4 3
-1 0 2
5
1
0
8
1
7
3
-3
-11
42
16
61
50
-6
We start with this row and column, and sum the products of each pair.
(1 x 5) + (0 x 1) + (3 x 0) + (-2 x 8) = -11
Further Example
June 2012 Paper 1 Q2
=
10
?
17
Test Your Understanding
Now you have a go...
a
If 𝐴 =
b
1
2
c
1
3
1
1
1
0
2
4
2
0
0
,𝐵 =
3
1
1
𝟎 ?𝟏
, 𝐴𝐵 =
2
𝟑 𝟑
𝟐 ?
3
=
𝟔
−1
7 10
=
?
15 22
N
N
N
1
1 2 3 2 = 14 ?
3
1
1 2 3
2 1 2 3 = 2 4? 6
3
3 6 9
?
Bro Exam Note: In IGCSEFM, you
will only have to multiply either a
2 × 2 by 2 × 1 or 2 × 2 by 2 × 1.
Identity Matrix
1 0
𝑎
Let 𝑰 =
and 𝐴 =
0 1
𝑐
Determine:
𝒂
𝒄
𝒂
𝐼𝐴 =
𝒄
𝐴𝐼 =
𝑏
.
𝑑
𝒃
?𝒅
𝒃
?𝒅
1 0
is known as the ‘identity matrix’.
0 1
Multiplying by it has no effect, i.e. 𝐴𝐼 = 𝐼𝐴 = 𝐴 for any matrix 𝐴.
𝑰 =
It may seem pointless to have such a matrix, but it’ll have more
importance when we consider matrices as ‘transformations’ later.
Although admittedly you won’t quite fully appreciate why we have it
unless you do Further Maths A Level…
Exercise 1
1
?
?
?
?
?
?
2
?
?
?
?
?
?
3
?
?
?
?
?
?
Exercise 1
4
?
?
?
?
?
?
5
?
?
?
?
?
?
6
?
?
?
?
?
?
Exercise 1
7
?
?
?
?
?
?
Harder Multiplication Questions
Matrix multiplications may give us simultaneous equations, which we solve in
the usual way.
June 2013 Paper 2 Q12
𝑥 2 − 12
4𝑥
=
𝑥 − 4𝑦
8
𝑥 2 − 12 = 4𝑥
𝑥?2 − 4𝑥 − 12 = 0
𝑥 = −2 𝑜𝑟 𝑥 = 6
𝑦 = −2.5 𝑜𝑟 𝑦 = −0.5
Test Your Understanding
AQA Worksheet 2
𝟐 + 𝟐𝒂 = 𝟏𝟐 → 𝒂 = 𝟓
𝟔 + 𝒂𝒃 = 𝟐𝟔
→ 𝒃=𝟒
?→ 𝒄 = 𝟓
𝟑+𝟐=𝒄
22
 2 a 
3
   
94 3 
7
Exercise 1b
4
1
Set 4 Paper 1 Q17
−𝟔 + 𝟕𝒂
= 𝟐𝟐
?
𝒂=𝟒
2
June 2013 Paper 2 Q11
−𝒂 𝟐𝒃 − 𝒄
𝟏
𝟎
𝒃
𝟑
?
𝟐𝒂 + 𝒂𝒃 = −𝟏
𝒂 − 𝟑𝒃 = 𝟐
𝒂 = 𝟐 + 𝟑𝒃
∴ 𝟐(𝟐 + 𝟑𝒃) + 𝟐 + 𝟑𝒃 𝒃 = −𝟏
𝟑𝒃𝟐 + 𝟖𝒃 + 𝟓 = 𝟎
𝟑𝒃 + 𝟓 𝒃 + 𝟏 = 𝟎
𝟓
𝒃 = −𝟏 𝒐𝒓 𝒃 = −
𝟑
𝒂 = −𝟏 𝒐𝒓 𝒂 = −𝟑
?
?
𝒂 = −𝟏, 𝒃 = 𝟑, 𝒄 = 𝟔
3
Set 2 Paper 2 Q16
𝟐
𝟓
𝒂 𝒂+𝒃
𝟐+𝒂 𝟑+𝒃
𝑸𝑷 =
𝒂
𝒃
∴ 𝒂 = 𝟎, 𝒃 = 𝟐
𝑷𝑸 =
?
Matrices representing transformations
Matrices can represent transformations to points in 2D or 3D space.
𝑥
Let us represent a point as the vector 𝑦
We can multiply it by a matrix:
𝟐𝒙
2 0 𝑥
= ?
𝑦
𝟐𝒚
0 2
(Note: You’re used to representing
points as coordinates like 𝑥, 𝑦 rather
than vectors, but it allows us to apply
matrix transformations to them more
easily in this form)
𝑦
Important Note: When we multiply by a matrix, it goes on the front, not after.
This is a bit like how with composite functions, e.g. 𝑔𝑓(𝑥), we applied 𝑓 to 𝑥
followed 𝑔. We go right to left.
What ‘transformation’ therefore does the
2 0
matrix
represent?
0 2
An enlargement by scale factor 2 about
?
the origin.
𝑥
𝑦
2𝑥
2𝑦
𝑥
A further example
What transformation does the matrix
0
−1
1
0
0 1
represent?
−1 0
𝑥
𝒚
?
𝑦 = −𝒙
Step 1: Find the effect
𝑥
on a point 𝑦 .
Step 2: Draw the old and
new point (using a specific
example point if you wish) to
see the effect.
𝑦
3
1
?
𝑥
1
−3
Transformation:
Rotation 𝟗𝟎° clockwise about
?
the origin.
Investigate
In pairs or otherwise, determine the transformations that each of these
matrices represents.
−1
0
0
1
Reflection in the
line 𝒙 = ?
𝟎
0
1
1
0
0
−1
Reflection in the
line 𝒚 =?𝟎
0
−1
Rotation 𝟏𝟖𝟎°
? origin.
about the
1
0
Reflection in the line
𝒚=𝒙 ?
−1
0
0
1
−1
0
Rotation 𝟗𝟎°
? about
anticlockwise
the origin.
0
−1
1
0
Rotation 𝟗𝟎°
clockwise ?
about the
origin.
1
0
0
1
No effect!
?
Going backwards
Work out the transformation that transforms a point 270° clockwise
about the origin.
𝑦
−1
3
?
?
3
𝑥1
Use a specific point or 𝑦
and find the effect of the
transformation.
𝑥
Work
0 out
−1
what
1 matrix
0
would have
this effect
𝑥
−𝑦
𝑦 = 𝑥
Transforming the unit square
Set 3 Paper 2 Q17
For more complex transformations
it’s not sufficient to look at the
effect on just one point: we can’t
fully see what the matrix is doing.
If we look at the effect on a unit
square (with coordinates
0,0 , 1,0 , 1,1 , 0,1 ), we can
better see the effect of a matrix
transformation on a region in the
𝑥-𝑦 plane.
Just apply the transformation to
each point of the unit square.
3
0
3
0
3
0
3
0
0
3
0
3
0
3
0
3
0
0
1
0
0
1
1
1
𝟎
?
𝟎
𝟑
= ?
𝟎
𝟎
= ?
𝟑
𝟑
= ?
𝟑
=
Test Your Understanding
Set 1 Paper 1 Q14
𝐴′
𝐵′
𝐶′
0
−1
0
−1
0
−1
0
−1
−1
0
−1
0
−1
0
−1
0
0
0
1
? 00
1
1
1
𝟎
𝟎
𝟎
=
−𝟏
−𝟏
=
𝟎
−𝟏
=
−𝟏
=
Exercise 3
1 [Jan 2013 Paper 2 Q15] Describe fully the
single transformation represented by the
0 −1
matrix
1 0
Rotation 𝟗𝟎° anticlockwise about the origin.
5
?
2 [Set 2 Paper 1 Q4] The transformation matrix
𝑎 2
maps the point 3,4 onto the point
−1 1
2, 𝑏 . Work out the values of 𝑎 and 𝑏.
𝒂 =?−𝟐, 𝒃 = 𝟏
?
6
𝑎
𝑏
3 [Set 3 Paper 1 Q6] The matrix −𝑎 2𝑏 maps
the point 5,4 onto the point 1,17 . Work
7
out the values of 𝑎 and 𝑏.
𝒂 = ?−𝟏, 𝒃 = 𝟏. 𝟓
4
[Worksheet 2 Q5] Work out the image of the
point D (1, 2) after transformation by the
𝟒
2 3
matrix
Solution:
? 𝟑
−1 1
[Worksheet 2 Q6] The point A(m, n) is
transformed to the point A (2, 0) by
2 3
the matrix
1 1
Work out the values of m and n.
𝒎 = 𝟐, 𝒏 = −𝟐
[Worksheet 2 Q8] Describe fully the
transformation given by the matrix
0 −1
−1 0
Reflection in the line 𝒚 = −𝒙
?
[Worksheet 2 Q9] The unit square
OABC is transformed by the matrix
ℎ 0
to the square OABC.
0 ℎ
The area of OABC is 27. Work out
the exact value of h.
𝒉 =?𝟑 𝟑
Combined Transformations
𝐴
𝑃
𝐵
?
𝐴𝑃
?
𝐵𝐴𝑃
𝐵𝐴
?
If a point 𝑃 is transformed by the matrix 𝐴 followed by the matrix
𝐵, what calculation would get the new point?
Therefore what matrix represents the combined transformation
of 𝐴 followed by 𝐵?
 The matrix 𝐵𝐴 represents the combined transformation
of 𝐴 followed by 𝐵.
Example
−1 0
A point 𝑃 is transformed using the matrix
, i.e. a reflection in the line 𝑥 = 0,
0 1
0 1
followed by
, i.e. a reflection in the line 𝑦 = 𝑥.
1 0
(a) Give a single matrix which represents the combined transformation.
(b) Describe geometrically the single transformation this matrix represents.
a
b
𝟎 𝟏
𝟏 𝟎
−𝟏 𝟎
𝟎 𝟏
=
?
𝟎 𝟏
−𝟏 𝟎
Rotation 𝟗𝟎° clockwise about the origin.
?
Test Your Understanding
Worksheet 2 Q7
Bro Note: The default direction of rotation is
anticlockwise if not specified.
0 ?1
1 0
0 1
𝑩=
−1? 0
0 1 0 1
1 0
𝑩𝑨 =
=
−1 0 1 0
0 −1 ?
(The question does not ask, but this represents a reflection in the line 𝑦 = 0)
𝑨=
Exercise 3
1 Point 3, −2 is transformed by the matrix
−1
followed by a further transformation
1
0 2
by the matrix
.
1 0
(i) Work out the matrix for the combined
𝟎 𝟐
transformation.
Solution:
𝟏 −𝟏
(ii) Work out the co-ordinates of the image point
of 𝑃.
Solution: (−𝟒, 𝟓)
3
1
0
The unit square is reflected in the 𝑥-axis
followed by a rotation through 180° centre
the origin. Work out the matrix for the
combined transformation.
Solution:
?
?
2 Point −1,4 is transformed by the matrix
3
−2
−1
followed by a further
2
1 0
transformation by the matrix
.
3 −2
(i) Work out the matrix for the combined
𝟑 −𝟏
transformation.
Solution:
𝟏𝟑 −𝟕
(ii) Work out the co-ordinates of the image point
of 𝑊.
Solution: (−𝟕, −𝟒𝟏)
?
?
4
?−𝟏
𝟎
𝟎
𝟏
The unit square is enlarged, centre the
origin, scale factor 2 followed by a
reflection in the line 𝑦 = 𝑥. Work out the
matrix for the combined transformation.
Solution:
?𝟎𝟐
𝟐
𝟎
Exercise 3
5 [Jan 2013 Paper 2 Q17] −1 0 represents a
0
1
0 1
represents a
1 0
reflection in the line 𝑦 = 𝑥.
Work out the matrix that represents a
reflection in the 𝑦-axis followed by a reflection
in the line 𝑦 = 𝑥.
𝟎 𝟏 −𝟏 𝟎
𝟎 𝟏
=
𝟏 𝟎
𝟎 𝟏
−𝟏 𝟎
reflection in the 𝑦-axis.
?
6
[June 2012 Paper Q22] The transformation
0 −1
matrix
maps a point 𝑃 to 𝑄. The
−1 0
1 0
transformation matrix
maps point 𝑄
0 −1
to point 𝑅.
Point 𝑅 is −4,3 . Work out the coordinates of
point 𝑃.
𝟏 𝟎
𝟎 −𝟏
𝟎 −𝟏
=
𝟎 −𝟏 −𝟏 𝟎
𝟏 𝟎
This is a rotation 𝟗𝟎° anticlockwise. So
𝟑
original point 𝑷 is
𝟒
?
7 [Set 1 Paper Q14b] The unit square
OABC is transformed by reflection in
the line 𝑦 = 𝑥 followed by
enlargement about the origin with
scale factor 2. What is the matrix of
the combined transformation?
𝟐 𝟎 𝟎 𝟏
𝟎 𝟐
=
?
𝟎 𝟐 𝟏 𝟎
𝟐 𝟎
3 0
−1 0
and 𝐵 =
.
0 3
0 1
The point 𝑃 2,7 is transformed by
matrix 𝐵𝐴 to 𝑃′. Show that 𝑃′ lies on
the line
7𝑥 + 2𝑦 = 0.
𝐴=
−𝟏
𝟎
𝟎 𝟑 𝟎 𝟐
−𝟔
=
𝟏 𝟎 ?
𝟑 𝟕
𝟐𝟏
𝟕 −𝟔 + 𝟐 𝟐𝟏 = 𝟎