MATH1002: Notes
Table of Contents
Module 1 ... 4
Scalars ... 4
Vectors ... 4
Vector Addition ... 4
Scalar Multiplication ... 5
Vector Addition and Scalar Multiplication in ℝ" ... 5
Vector Algebra in ℝ# ... 5
Properties of Vector Algebra ... 5
Module 2 ... 6
Linear Combinations of Vectors ... 6
Dot Product ... 6
Dot Product & Length ... 6
Cauchy- Schwartz Inequality ... 7
Triangle Inequality ... 7
Unit Vectors ... 7
Distance and Angles between Vectors ... 7
Projections ... 8
Module 3 ... 9
Cross Product ... 9
Geometric Meaning of the Cross Product: Direction ... 9
Geometric Meaning of the Cross Product: Length ... 9
Lines in ℝ$: Normal Form and General Form ... 10
Lines in ℝ$ and ℝ": Vector Form ... 10
Parametric Form for Lines in ℝ$ ... 11
Find a vector form and parametric equations for the line passing through P and Q ... Error! Bookmark not defined. Parametric Form for Lines in ℝ" ... 11
Lines in ℝ: Overview and Comparison ... 11
Module 4 ... 12
Normal Planes in ℝ" ... 12
Planes in ℝ": Vector Form and Parametric Equations ... 12
Planes in ℝ": Comparing Normal and Vector Form ... 13
Span and Linear Independence ... 13
Systems of Linear Equations ... 13
Representing a System of Linear Equations as an Augmented Matrix ... 14
Module 5 ... 16
Elementary Row Operations ... 16
Row Echelon Form and Back-Solving ... 16
Solving Systems of Linear Equations Using Gaussian Elimination ... 17
Gaussian Elimination Process ... 17
Reduced Row Echelon Form and Gauss-Jordan Elimination ... 18
Introduction to Matrices ... 18
Module 6 ... 19
Addition & Scalar Multiplication of Matrices ... 19
Matrix Multiplication ... 19
Properties of Matrix Multiplication ... 20
Matrix Algebra: Scalar Multiplication, Matrix Addition & Matrix Multiplication . Error! Bookmark not defined. Matrix Transposition ... 20
Module 7 ... 21
Matrix Inverses ... 21
Determinants for $ × $ Matrices ... 23
Application of Inverses: Solving Systems of Linear Equations ... 23
Module 8 ... 24
Elementary Matrix ... 24
Determinants ... 24
Module 9 ... 25
Determinants and EROs ... 25
Using EROs to Calculate Determinants ... 26
Determinants and Inverses ... 26
Eigenvalues & Eigenvectors ... 27
Module 10 ... 28
Trace ... 28
Vector Spaces, Span & Linear Independence ... 28
Eigenspaces ... 29
Application of Eigen-things to Matrix Computations ... 29
Diagonalisation ... 31
Diagonalisability & Geometric Multiplicity ... 32
Powers of Diagonalisable Matrices ... 32
Leslie Population Model ... 33
Module 12 ... 34
Probability Vectors & Stochastic Matrices ... 34
Markov Chains/Processes ... 34
Steady State Vector ... 35
Regular Markov Chains ... 35
Module 1 Scalars
Real numbers are called scalars → denoted by ℝ Vectors
Vectors are quantities that are characterised by their magnitude (length) and direction (not their position)
•
Vectors whose tail is at the origin are called “in standard position”
•
Set of all vectors in the 2D plane is denoted by ℝ
!(3D plane - ℝ
") Algebraic Notation
•
If # = (&
#, &
!) is a point in the plane, we consider the vector from the origin ) = (0,0) to #
o
We write )# +++++⃗ = - &
#&
!.
§
&
#in the /-direction
§
&
!in the 0-direction
o)# +++++⃗ is the position vector of #
o
The scalars &
#, &
!are the components of )# +++++⃗
•
If # = (&
#, &
!) and 1 = (2
#, 2
!) then we consider the arrow from # (tail) to 1 (head)
o#1 +++++⃗ = 32
#− &
#2
!− &
!5
§
The displacement vector from # to 1
•
Two vectors - &
#&
!. and 3 2
#2
!5 are equal ⟺ they have the same components (&
#= 2
#, &
!= 2
!)
Vector Addition
•
Geometric Definition: Given two vectors # +++⃗ and 1 +++⃗ , we produce a new vector # +++⃗ + 1 +++⃗
by ‘adding head to tail’
o
New arrow # +++⃗ + 1 +++⃗ goes from tail of # +++⃗ to head of 1 +++⃗
•
Algebraic Definition: The vector sum of # +++⃗ = - &
#&
!. and 1 +++⃗ = 32 2
#!5 is given by:
+++⃗ + 1 # +++⃗ = 3&
#+ 2
#&
!+ 2
!5
•
Parallelogram Rule (Commutativity): # +++⃗ + 1 +++⃗ = 1 +++⃗ + # +++⃗
•
Associative Law of Vector Addition (Associativity): 9# +++⃗ + 1 +++⃗: + ; +++⃗ = # +++⃗ + 91 +++⃗ + ; +++⃗:
Scalar Multiplication
Take a scalar < ∈ ℝ and a vector # +++⃗ and produce a new vector < × # +++⃗ = <# +++⃗
•
Geometric Definition: <# +++⃗ is the vector that we get by scaling # +++⃗ until it has length
|<| times its original length
o
If < > 0, <# +++⃗ must point in the same direction as # +++⃗
o
If < < 0, <# +++⃗ must point in the opposite direction as # +++⃗
o
If < = 0, then <# +++⃗ = 0 +++⃗
•
Algebraic Definition: If # +++⃗ = - &
#&
!., then <# +++⃗ = - <&
#<&
!.
•
Negative: −1 × # +++⃗ = −# +++⃗
•
Vector Subtraction: # +++⃗ − 1 +++⃗ ≔ # +++⃗ + 9−1 +++⃗:
Vector Addition and Scalar Multiplication in ℝ
$Addition: D
&
#&
!&
"E + D 2
#2
!2
"E = D
&
#+ 2
#&
!+ 2
!&
"+ 2
"E Multiplication: < × D
&
#&
!&
"E = D
<&
#<&
!<&
"E Vector Algebra in ℝ
%For F a positive natural number, a vector in ℝ
&in component form is given by an F-tuple of real numbers: &⃗ = D
&
#⋮
&
&E
The entries &
'are called the components of &⃗
•
Two vectors are equal if their components are equal: &
'= 2
'HIJ K = 1,2, … , F Properties of Vector Algebra
THEOREM: Let N+⃗, O⃗, P ++⃗ be vectors in ℝ
&, and take <, Q ∈ ℝ:
•
N+⃗ + O⃗ = O⃗ + N+⃗ (Commutativity)
•
(N+⃗ + O⃗) + P ++⃗ = N+⃗ + (O⃗ + P ++⃗) (Assosciativity)
•
N+⃗ + 0+⃗ = N+⃗
•
N+⃗ + (−N+⃗) = 0+⃗
•
<(N+⃗ + O⃗) = <N+⃗ + <O⃗ (Distributivity)
•
(< + Q)N+⃗ = <N+⃗ + QN+⃗ (Distributivity)
•
<(QN+⃗) = (<Q)N+⃗
•
1N+⃗ = N+⃗
