Vector Spaces
Linear Transformations
Eigenvalues and Eigenvectors
Probability and Statistics
Supplementary
UNSW Mathematics Society presents
MATH1231/1241 Revision Seminar
(Higher) Mathematics 1B
Algebra
Presented by Joanna Lin and Karen Zhang
T2, 2020

Vector Spaces
Linear Transformations
Eigenvalues and Eigenvectors
Probability and Statistics
Supplementary
Table of Contents I
1
Vector Spaces
Vector Spaces
Subspaces
Linear Combinations and Spans
Linear Dependence
2
Linear Transformations
Definition of Linear Transformation
Linear Maps and Matrices
Kernel of a map
Image and Rank
3
Eigenvalues and Eigenvectors
Definition
Computation of eigenvalues and eigenvectors
Diagonalisation
Applications
2 / 143
MATH1231/1241 Revision Seminar
Presented by: Joanna Lin and Karen Zhang

Vector Spaces
Linear Transformations
Eigenvalues and Eigenvectors
Probability and Statistics
Supplementary
Table of Contents II
4
Probability and Statistics
Probability
Conditional Probability
Random Variables
Cumulative Distribution Function
Probability Distribution
Expected Value and Variance
Binomial Distribution
Probability Density Function
Normal Distribution
[X] Exponential Distribution
5
Supplementary
More Theory
Geometric Representations of Linear Maps
Questions
3 / 143
MATH1231/1241 Revision Seminar
Presented by: Joanna Lin and Karen Zhang

Vector Spaces
Linear Transformations
Eigenvalues and Eigenvectors
Probability and Statistics
Supplementary
Vector Spaces
4 / 143
MATH1231/1241 Revision Seminar
Presented by: Joanna Lin and Karen Zhang

Vector Spaces
Linear Transformations
Eigenvalues and Eigenvectors
Probability and Statistics
Supplementary
Vector Spaces
Vector Space Definition
In the following definition, we assume that
u
,
v
and
w
are elements
of a particular vector space.
λ
and
μ
are scalars in the field over
which the vector space is defined.
A vector space
V
over a field
F
is a non-empty set in which
addition of vectors and scalar multiplication are defined in such a
way that the following axioms are satisfied:
1
Closure under Addition
.
2
Associative Law of Addition
: (
u
+
v
) +
w
=
u
+ (
v
+
w
).
3
Commutative Law of Addition
:
u
+
v
=
v
+
u
.
4
Existence of a Zero
: There exists a
0
∈
V
such that
u
+
0
=
u
.
5
Existence of a Negative
: For each
u
∈
V
, there exists a
v
∈
V
where
u
+
v
=
0
.
5 / 143
MATH1231/1241 Revision Seminar
Presented by: Joanna Lin and Karen Zhang