14
Quadratic Forms
Definition 14.1. A quadratic form in n variables is a function f : Rn R
of the form
X
f (x) = f (x1 , ., xn ) =
cij xi xj
()
1ijn
where x Rn and cij R(1 i j n). Alternatively, a quadratic form is
a homogeneous polynomial of degree 2 in

5
Vectors, Eigenvalues and Subspaces
5.1
Vectors and Matrices
Recall that real n-space is the set
Rn = cfw_(a1 , . . . , an ) | a1 , . . . , an R.
The elements of Rn are called n-vectors, or just vectors when n is clear.
The Zero Vector. We write 0n for t

Prerequisite Material
This course builds directly upon MATH10101 Sets, Numbers and Functions, from last semester.
You will need to understand many of the concepts from that course, including:
proofs (including by contradiction and induction);
sets (ways

2
Systems of Linear Equations
A system of equations of the form
x + 2y = 7
2x y = 4
or
5p 6q + r = 4
2p + 3q 5r = 7
6p q + 4r = 2
is called a system of linear equations.
Definition. An equation involving certain variables is linear if each side of the equ

10
Co-ordinates and change of bases
Let V be a vector space with dim(V ) = n and let B = cfw_v1 , ., vn be a
basis for V . Then every vector v V has a unique expression as a linear
combination
v = a1 v1 + . + an vn
for some a1 , ., an K. The scalarsa1 ,

13
Subspaces Associated with Matrices
Definition 13.1. Let A be an m n matrix in Mmn (K).
1. The row space of A is the subspace of K n spanned by the rows of A,
denoted by row(A).
2. The column space of A is the subspace of K m spanned by the columns
of A

4
Determinants
The determinant of a square matrix is a scalar (i.e. a number) which can be associated to it, and which
contains a surprisingly large amount of information about the matrix. To define it we need some extra
notation:
Definition. If M is an n

12
The Matrix of a Linear Transformation
Definition 12.1. Let V and W be finite dimensional vector spaces over K
with bases B = cfw_v1 , ., vn and C = cfw_w1 , ., wm respectively and let T : V
W be a linear transformation. Then the m n matrix whose col

11
Linear Transformations
Definition 11.1. A linear transformation from a vector space V (over
K) to a vector space W (over K) is a function T : V W such that for all
u, v V and all a K,
1. T (u + v) = T (u) + T (v);
2. T (au) = aT (u).
Note: It follows f

3
Elementary Matrices and Calculating Inverses
Definition 3.1. An n n matrix E is an elementary matrix if it can be obtained from the identity
matrix In by a single elementary row operation. For an elementary row operation we write E we write
E for the co

9
Bases
Definition 9.1. A subset B of a vector space V is a basis for V if B spans
V and B is linearly independent.
1. The standard basis for K n is the set cfw_e1 , ., en
0
.
.
where ei = 1 , where the 1 appears in the i-th row. There are
.
.

8
Spanning Sets and Linear Independence
Spanning Sets
Let V be a vector space over K. An expression of the form
a1 v1 + a2 v2 + . + am vm =
m
X
ai v i ,
i=1
where v1 , ., vm V and a1 , ., am K is called a linear combination of
the vectors v1 , ., vm .
Def

MATH10202: Linear Algebra A Part 2
7
Vector Spaces
Definition and Examples
Definition 7.1. A field is a set K with two binary operations + and ,
called addition and multiplication such that:
1.(i) a + (b + c) = (a + b) + c for all a, b, c K
(ii) a + b = b

MATH10202 Linear Algebra A 201516
These notes accompany the part of the course given by Mark Kambites.
What is Linear Algebra and Why Do I Need It?
Linear means to do with lines. Linear algebra is the algebra of linear equations, which are
equations whose

6
Orthogonality
6.1
The Euclidean Inner Product and Norm
Definition. Let u = (a1 , . . . an ), v = (b1 , . . . , bn ) Rn . The (Euclidean15 ) inner product (also known
as the scalar product or dot product on Rn is defined by
u | v = a1 b1 + a2 b2 + + an b