Matrix Transformations Dr J Frost jfrosttiffinkingstonschuk wwwdrfrostmathscom Last modified 3 rd January 2016 The specification Introduction A matrix plural matrices is simply an array of numbers ID: 594438
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Slide1
IGCSE FM Matrix Transformations
Dr J Frost (jfrost@tiffin.kingston.sch.uk)www.drfrostmaths.com
Last modified:
3
rd January 2016
The specification:Slide2
Introduction
A matrix (plural: matrices) is
simply an ‘array’ of numbers
, e.g.
For the purposes of IGCSE Further Maths, you should understand matrices as
a way to transform points
.
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
.
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. Slide3
(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.
* It really isn’t.
?
In an
‘adjacency matrix’,
the number in the
i
th
row and j
th column is the number of edges directly connecting node (i.e. dot) i to dot j
?Slide4
Using matrices to represent data
In my 4
th
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.Slide5
ζ
Matrix Fundamentals
Matrix Algebra
Understand the dimensions of a matrix, and operations on matrices, such as addition, scalar multiplication and matrix multiplication.Slide6
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
?
?Slide7
Matrix Fundamentals
#2 Notation/Names for Matrices
A matrix can have square or curvy brackets*.
* The textbook only uses curvy.
Matrix
Column Vector
(The vector you know
and love)
Row Vector
So a matrix with one column is simply a vector in the usual sense.Slide8
Matrix Fundamentals
#3 Variables for Matrices
If we wish a variable to represent a matrix, we use bold, capital letters.
Slide9
Matrix Fundamentals
#4
Adding/Subtracting Matrices
Simply add/subtract the corresponding elements of each matrix.
They must be of the same dimension.
?
?Slide10
Matrix Fundamentals
#5
Scalar Multiplication
A scalar is a number which can ‘scale’ the elements inside a matrix/vector.
?
?
?
1
2
3Slide11
Matrix Fundamentals
#6
Matrix Multiplication
This is where things get slightly more complicated...
1 0 3 -2
2 8 4 3
7 -1 0 2
5 1
1 7
0 3
8 -3
-11
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
16
Now repeat for the next row of the left matrix...
42
61
50
-6Slide12
Further Example
June 2012 Paper 1 Q2
?Slide13
Test Your Understanding
Now you have a go...
?
?
?
a
b
c
N
N
N
Bro Exam Note
: In IGCSEFM, you will only have to multiply either a
by
or
by
.
If
?
?
?Slide14
Identity Matrix
Let
and
.
Determine:
is known as the ‘identity matrix’.
Multiplying by it has no effect, i.e.
for any matrix
.
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…
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?Slide15
Exercise 1
1
2
3
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?Slide16
Exercise 1
4
5
6
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?Slide17
Exercise 1
7
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?Slide18
Harder Multiplication Questions
June 2013 Paper 2 Q12
?
Matrix multiplications may give us
simultaneous equations
, which we solve in the usual way.Slide19
Test Your Understanding
AQA Worksheet 2
?Slide20
Exercise 1b
June 2013 Paper 2 Q11
Set 2 Paper 2 Q16
Set 4 Paper 1 Q17
1
2
3
4
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?
?Slide21
Matrices representing transformations
Matrices can represent transformations to points in 2D or 3D space.
Let us represent a point as the vector
We can multiply it by a matrix:
(Note
: You’re used to representing points as coordinates like
rather than vectors, but it allows us to apply matrix transformations to them more easily in this form)
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.
, we applied
to
followed
. We go right to left.
What ‘transformation’ therefore does the matrix
represent?
An enlargement by scale factor 2 about the origin.
?
?Slide22
A further example
What transformation does the matrix
represent?
Step 1
: Find the effect on a point
.
Step 2
: Draw the old and new point (using a specific example point if you wish) to see the effect.
Transformation:
Rotation
clockwise about the origin.
?
?
?Slide23
Investigate
In pairs or otherwise, determine the transformations that each of these matrices represents.
Rotation
clockwise about the origin.
Rotation
anticlockwise
about the origin.
Reflection in the line
No effect!
Rotation
about the origin.
Reflection in the line
Reflection in the line
?
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?Slide24
Going backwards
Work out the transformation that transforms a point
clockwise about the origin.
?
Use a specific point or
and find the effect of the transformation.
?
Work out what matrix would have this effectSlide25
Transformin
g 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
), we can better see the effect of a matrix transformation on a region in the
-
plane.
Just apply the transformation to each point of the unit square.
?
?
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?Slide26
Test Your Understanding
Set 1 Paper 1 Q14
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Slide27
Exercise 3
[Jan 2013 Paper 2 Q15] Describe fully the
single
transformation represented by the matrix
[
Set 2 Paper 1 Q4] The transformation matrix
maps the point
onto the point
. Work out the values of
and
.
[
Set 3 Paper 1 Q6] The matrix
maps
the point
onto the point
. Work out the values of
and
.
[
Worksheet 2 Q5]
Work out the image of the point
D
(
1, 2) after transformation by the matrix
Solution:
[Worksheet 2 Q6] The point
A
(
m
,
n
) is transformed to the point
A
(
2, 0) by the matrix
Work out the values of
m
and
n
.
[
Worksheet 2 Q8] Describe fully the transformation given by the matrix
Reflection in the line
[
Worksheet 2 Q9] The unit square
OABC
is transformed by the matrix
to the square
OA
B
C
.
The area of
OA
B
C
is 27.
Work out the exact value of
h
.
1
2
3
4
5
6
7
?
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?
?
?Slide28
Combined Transformations
If a point
is transformed by the matrix
followed by the matrix
, what calculation would get the new point?
Therefore what matrix represents the combined transformation of
followed by
?
?
?
?
!
The matrix
represents the combined transformation of
followed by
.
Slide29
Example
A point
is transformed using the matrix
, i.e. a reflection in the line
, followed by
, i.e. a reflection in the line
.
Give a single matrix which represents the combined transformation.
Describe geometrically the single transformation this matrix represents.
Rotation
clockwise about the origin.
a
b
?
?Slide30
Test Your Understanding
Worksheet 2 Q7
Bro Note
: The default direction of rotation is
anticlockwise
if not specified.
(The question does not ask, but this represents a reflection in the line
)
?
?
?Slide31
Exercise 3
Point
is transformed by the matrix
followed by a further transformation by the matrix
.
(
i
) Work out the matrix for the combined transformation
.
Solution:
(ii) Work out the co-ordinates of the image point of
.
Solution:
Point
is transformed by the matrix
followed by a further transformation by the matrix
.
(
i
) Work out the matrix for the combined transformation
.
Solution
:
(ii) Work out the co-ordinates of the image point of
.
Solution:
The
unit square is reflected in the
-axis followed by a rotation through
centre the origin.
Work out the matrix for the combined transformation.
Solution:
The unit square is enlarged, centre the origin, scale factor 2 followed by a reflection in the line
. Work out the matrix for the combined transformation.
Solution:
1
3
2
4
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?Slide32
[Jan 2013 Paper 2 Q17]
represents a reflection in the
-axis.
represents a reflection in the line
.
Work out the matrix that represents a
reflection
in the
-axis followed by a reflection in the line
.
[June 2012 Paper Q22] The transformation matrix
maps a point
to
. The transformation matrix
maps point
to point
.
Point
is
. Work out the coordinates of point
.
This is a rotation
anticlockwise. So original point
is
Exercise 3
[Set 1 Paper Q14b] The unit square OABC is transformed by reflection in the line
followed by enlargement about the origin with scale factor 2.
What is the matrix of the combined transformation
?
and
.
The point
is transformed by matrix
to
. Show that
lies on the line
.
5
7
6
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