Dr J Frost jfrosttiffinkingstonschuk Last modified 29 th August 2015 Introduction A matrix plural matrices is simply an array of numbers eg But the power of matrices comes from being able to multiply matrices by vectors and matrices by matrices and invert them ID: 702618
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Slide1
FP1: Chapter 4 Matrix Algebra
Dr J Frost (jfrost@tiffin.kingston.sch.uk)
Last modified:
29
th
August 2015Slide2
Introduction
A matrix (plural: matrices) is
simply an ‘array’ of numbers
, e.g.
But the power of matrices comes from being able to multiply matrices by vectors and matrices by matrices and ‘invert’ them: we can:
represent
linear transformations
using matrices (e.g. rotations, reflections and enlargements)
Use them to
solve linear simultaneous equations.
The first of these means matrices are particularly useful in 3D graphics/animation, since they allow us to rotate the camera and project 3D data onto a 2D viewing plane. But matrices are used everywhere, including robotics, computer vision, optimisation, classical and quantum mechanics, electromagnetism, optics, graph theory, statistics, ...
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
.Slide3
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
Using matrices to represent data
In S1, you learnt how we could extend the idea of
variance
(i.e. how much a variable varies) to that of
covariance
(i.e. how much two variables vary with each other).
A
covariance matrix
allows us to store the covariance between variables all in one convenient structure.
This allows us for example to extend a Normal Distribution from one dimension (i.e. involving just one variable) to multiple dimensions.
1D
2DSlide6
ζ
Matrix Fundamentals
Matrix Algebra
Understand the dimensions of a matrix, and operations on matrices, such as addition, scalar multiplication and matrix multiplication.Slide7
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
?
?Slide8
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.Slide9
Matrix Fundamentals
#3 Variables for Matrices
If we wish a variable to represent a matrix, we use bold, capital letters.
Slide10
Matrix Fundamentals
#4 Elements
Each value within a matrix is known as an
element
.
If
A
is a matrix, then we can refer to the element in the
i
th
row and jth column as
.
Note that you do not need to know this notation for referencing elements for the purposes of FP exams, but it’s worthwhile knowing.
?
?
?Slide11
Matrix Fundamentals
#5 Adding/Subtracting Matrices
Simply add/subtract the corresponding elements of each matrix.
They must be of the same dimension.
?
?Slide12
Matrix Fundamentals
#6 Scalar Multiplication
A scalar is a number which can ‘scale’ the elements inside a matrix/vector.
?
?
?
1
2
3Slide13
Matrix Fundamentals
#7 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
You will see in C4 that this is known as finding the “dot/scalar product” of the two vectors.
16
Now repeat for the next row of the left matrix...
42
61
50
6Slide14
=
Matrix Fundamentals
#7 Matrix Multiplication
Now you have a go...
?
?
?
?
?
?
a
b
c
d
e
fSlide15
Matrix Fundamentals
#7 Matrix Multiplication
Matrix multiplications are not always valid: the dimensions have to agree.
Note that only
square matrices
(i.e. same width as height) can be raised to a power.
Dimensions of A
Dimension of B
Dimensions
of AB (if valid)
2
3
3
4
2
4
1
32 3Not valid.
6
2
2
4
6
4
1
3
3
1
1
1
7
5
7
5
Not
valid.
10
10
10
9
10
9
3
3
3
3
3
3
?
?
?
?
?
?
?Slide16
Exercise 4C
Q1, 3, 5, 6, 8, 9Slide17
ζ
Linear Transformations
Matrix Algebra
Appreciate what linear transformations are, and how we can use matrices to represent them.Slide18
Position vectors
You should be familiar with the difference between a vector and a point:
Point:
represents a position in space.
Vector:
represents a movement.
A position vector allows us to treat a point as a vector.
A position vector [x y] represents the movement from the origin (0,0) to the point (
x,y
).
?
?
x
y
Point (5,3)
Position vector [5 3]
For the remainder of the chapter, when I say ‘point’, I really mean ‘position vector’.Slide19
Linear Functions
From GCSE, you are used to a linear function looking like the equation of a straight line:
More generally, if we had multiple inputs, a linear function is:
When the inputs and outputs of the function are
vectors
however, then we have a
linear map/transformation
if:
The first condition ensures that if we scale the original point x, then this scales the transformed point too. The second says that if we add two vectors, then resulting vector will be the same then if we transformed each of the vectors individually then added them – we say that the transformation ‘preserves vector addition and scalar multiplication’.
and
represent vectors here not the
and
value of a vector.
Slide20
Linear Transformations
This follows by letting the scalar
in
This means that the origin is unaffected by a linear transformation.
Suppose that
0
is the
0vector
, i.e. The position vector (0,0) in 2D, (0,0,0) in 3D, etc.
Then for a linear transformation f:
?
Can you express the following transformations as functions?
If linear, prove it, otherwise find a counterexample.
Example 1
Reflects a 2D point in the yaxis.
Proof of linearity:
?
?Slide21
Linear Transformations
Example 2
Rotates a 2D point 90
clockwise about the origin.
Linear?
Yes (proof similar to before)
?
?
Example 3
Enlarges by a scale factor of 3 with the origin as the centre of enlargement.
Linear?
Yes. Proof:
?
?Slide22
Linear Transformations
Example 4
Projects a 3D coordinate onto the
xy
plane, almost as if you were observing a 3D point where your viewing window/eyes are the
xy
plane.
Linear?
Yes (proof left as exercise)
?
?
This is useful in 3D graphics, because you need to turn 3D data into a 2D image by projecting onto a viewing plane.
Example 5
Translate a 2D point 1 unit to the right.
?
Linear?
No! The origin is not preserved.
?Slide23
Linear Transformations
As we have seen, the following transformations are linear and not linear:
Reflection
(where line of reflection goes through the origin)
Rotation
(with centre at the origin)
Enlargement
(with centre at the origin)
Linear
NonLinear
Translation.
!
A linear transformation in general is when each output is a linear combination of the inputs
,
e.g
:
While a transformation can create or destroy dimensions, in FP1 we’ll just be dealing which transformation which transform 2D points to 2D points. Slide24
Affine Transformations
(Not mentioned in FP1)
Affine transformations extend linear transformations by
allowing a constant term
. This hence allows us to represent translations.
Linear
Affine
Linear Transformations
The following examples are
NOT
linear transformations:
(can you prove it?)
Slide25
Using Matrices for Linear Transformations
Matrices allow us to represent all possible linear transformations.
Reflection in
axis:
?
Rotation 90
clockwise about the origin.
?Slide26
Using Matrices for Linear Transformations
Matrices allow us to represent all possible linear transformations.
?
?
?
?
?Slide27
Basis Vectors and Vector Spaces
(Not mentioned in FP1)
Basic vectors
are usually (but not necessarily) the unit vectors along each axis.
A
vector space
is all possible points that can be obtained by some linear combination of the basis vectors.
Example: 2D space
Basic vectors:
Vector space:
All position vectors
since
Intuitively this is obvious: any point in 2D space can be obtained by some movement along the
axis and some movement along the
axis.
Our basis vectors could have been any nonparallel vectors.
?
?Slide28
Using Matrices for Linear Transformations
There’s a groovy trick that allows us to work out what matrix to use for a particular transformation. Observe:
Thus, each column of the matrix tells us how the unit vectors are transformed. These transformed basis vectors form the new vector space.
x
y
x
y
Example: Reflection in
axis.
?
First column is first basis vector transformed.
?
?Slide29
Using Matrices for Linear Transformations
x
x
Rotation
anticlockwise about the origin.
?
x
x
Rotation
anticlockwise about the origin.
?
?
?Slide30
Rotation for multiples of
In the FP1 specification, you are only required to identify rotations for multiples of
, although the rotation matrix for
anticlockwise in general is in your formula booklet.
Rotation of
anticlockwise
Rotation of
clockwise
Rotation of
anticlockwise
Bro Tip
: To remember these without having to plug in your
in the above matrix, just remember the
scaling, and then just visually think how each basis vector is transformed to get the signs of the 1s correct.
?
?
?Slide31
Using Matrices for Linear Transformations
x
x
Enlargement by scale factor 2 about the origin.
?
x
x
No change.
?
?
?
This is known as the identity matrix, and is denoted by
.
Slide32
A quick digression…
An identity element in general is a special type of element on a set (e.g. the set of integers, or real numbers, or matrices) with respect to a binary operator on the set.
An element
is an identity element if for all
in the set and some operator
then:
Set
Operation
Identity Element
0 (since
)
1 (since
)
square matrices
Matrix multiplication
(since
)
(
e.g
, but you tend to just see
)
Set
s
(since
)
Boolean Algebra
(since
)
matrices
Matrix addition
The
“0matrix”
.
Vectors
The “0vector”
.
Set
Operation
Identity Element
Matrix multiplication
Set
s
Boolean Algebra
Matrix addition
Vectors
(Not mentioned in FP1)
?
?
?
?
?
?
?Slide33
Using Matrices for Linear Transformations
x
x
Reflection in the line
?
?
Since for any line of length
and inclination
, the
component is
and
component
, then gradient is consequently
, i.e.
must be the angle between the
axis and line of reflection.
We can see from the diagram above that when we reflect
we can form the following triangle:
1
Thus the vector gets transformed to
We can do something similar to transform
Slide34
Test Your Understanding
Jan 2011 Q2
June 2011 Q3
Reflections in the
axis.
Enlargement by scale factor 3 about the origin.
A reflection in the line
?
?
?Slide35
Exercise 4E
Describe the transformations represented by the matrix:
a)
Reflection in
axis.
b)
Rotation
anticlockwise
about
.
c)
Rotation
clockwise
about
.
And for these:
a)
Rotation
clockwise
about
.
b)
Enlargement scale
factor 4, centre
.
c)
Rotation
anticlockwise about
.
1
3
4
Find the matrix that represents these transformations.
Rotation of
clockwise about
.
Reflection in the
axis.
Enlargement centre
scale factor 2.
5
Find the matrix that represents these transformations.
Enlargement scale factor 4 centre (0,0).
Reflection in the line
.
Rotation about
of
clockwise.
?
?
?
?
?
?
?
?
?
?
?
?Slide36
Combined Transformations
We know that for a position vector
and a matrix
representing some transformation, then
is the transformed point.
If we wanted to apply a transformation represented by a matrix
followed by another represented by
, what transformation matrix do we use to represent the combined transformation?
This is because to apply the effect of
followed by
, we have:
(
because matrix multiplication is ‘associative’*)
Bro Tip: Ensure that you put these matrices in the right order – the first that gets applied is on the right!
* A binary operator
is
associative
if
, i.e. when we multiply matrices, the order in which we multiply them doesn’t matter.
?Slide37
Combined Transformations
Represent as a single matrix the transformation representing a reflection in the line
followed by a stretch on the
axis by a factor of 4.
=
?
?
?
Represent as a single matrix the transformation representing a rotation
anticlockwise
about the point
followed by a reflection in the line
.
=
?
?
?
What single transformation is this?
Reflection in the
axis.
?Slide38
Exercise 4F
Page 93
Q1, 3, 5, 7Slide39
Matrix Inverses
In maths, we are used to functions having an inverse.
It would seem logical to have some inverse of multiplying something by a matrix, so that we can represent ‘undoing’ the transformation (e.g. the inverse of the matrix representing
rotation clockwise would be a matrix representing a rotation
anticlockwise).
!
If
then
is the ‘inverse’ of
, so that if
,
since the effect of a transformation followed by its inverse has no effect.
Slide40
Determinants
The determinant of a matrix
is
So
If
, then
is a
singular matrix
and
does not exist
(since we can’t divide by 0).
If
, then
is a
nonsingular matrix
and
exists.
A
det
(A)
1
2
3
2
A
det
(A)
1
2
3
2
?
?
?
?
It’s worthwhile reflecting on the significance of the determinant: in the context of transformations, it ‘determines’ whether we can ‘undo’ a transformation. And later, we’ll see that it can tell us whether a series of simultaneous equations are solvable.Slide41
Practicing the Inverse
Divide by determinant.
Swap NWSE elements.
Make SWNE elements negative.
Click to
BroinverseSlide42
Test Your Understanding
For what value of
is
singular? Given
is not this value, find the inverse.
?
?
?
?
?Slide43
Using
A matrix multiplied by its inverse is the identity matrix.
If
and
are nonsingular matrices, prove that
If
then
The general strategy for solving these kinds of equations is to multiply whatever’s on the front of the multiplication by its inverse, to ‘cancel it out’.
?
If
and
are
nonsingular matrices such that
. Prove that
.
?Slide44
Exercise 4G
Page 97
Q1a, c, e
2a, c
4, 6, 8, 10, 11Slide45
Inverse matrices represent inverse transformations
!
Suppose
and
are column vectors. Then if
, then
.
The inverse matrix therefore allows us to retrieve the original point/position vector before a transformation.
The triangle
has vertices at
,
and
. The matrix
transforms
to the triangle
with vertices at
and
.
Sketch the two triangles, and hence show that
So the determinant was the scale factor of the area.

1 1
(2,4)
(4,10)
(4,3)
4 4
?Slide46
Area scale factor
We saw in this example that:
Area of image
Area of object
i.e
. the determinant tells us how the area is scaled under the transformation matrix
.
Area of Object
Transformation Matrix
Area of Image
Area of Object
Transformation Matrix
Area of Image
?
?
?
?Slide47
Exam Question
Edexcel FP1  Jan 2011
4
Note that we’re given the points on the
image
! So use inverse transformation.
So points are
.
?
?
?
?Slide48
Exercise 4I
The matrix
is used to transform the rectangle
with vertices at the points
,
,
and
.
a) Find the coordinates of the vertices of the image of
.
b) Calculate the area of the image of
.
A rectangle of area 5cm
2
is transformed by the matrix
. Find the area of the image of the rectangle when
is:
a)
c)
e)
The triangle
has area 6cm
2
and is transformed by the matrix
where
is a constant, into triangle
.
Find
in terms of
.
Given that the area of
is 36cm
2
, find the possible values of
.
1
4
5
Area = 40
a) 70 c) 15 e) 90
?
?
?Slide49
Frost Life
Stories
TM
In the game
Assassin’s Creed II
, you encounter a variety of
concentric ring picture puzzles
, which upon successfully completing, you unlock a segment of a secret video.
Rings are
connected in pairs
, and must be rotated together in their pairs. The aim is to form a complete picture. Different possible pairs can be selected, for example, where there just 3 rings, you could rotate A and B together, B and C together or C and A together.
Only certain pairings are available.
Because I’m a massive geek, I formed simultaneous equations and used a matrix inverse to solve them, which therefore told me how many times to rotate each pair.
We’ll see how we can do this.Slide50
Using Matrices For Simultaneous Equations
Turn the following simultaneous equations into a matrix:
Hence, use a matrix inverse to solve these simultaneous equations.
Let
. Then
So
and
.
?
?Slide51
Test your understanding
Solve (using matrices) the simultaneous equations:
So
,
.
Explain why the following simultaneous equations have no solution:
The matrix
has no inverse because the determinant is 0.
Q
Q
?
?Slide52
Exercise 4J
Use inverse matrices to solve the following simultaneous equations.
?
?
?
?
1a
1b
2a
2b