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Vectors
Two Operations

Many familiar sets have two operations.
• Sets: Z, R, 2  2 real matrices M2(R)
• Addition and multiplication
• All are groups under addition.

0 is a problem for multiplication.
• Without 0 the sets are groups
• Multiplication is not commutative for M2(R)
Rings and Fields

A ring is a set with two
operations (R, +, )

Rings are commutative
groups under addition.

Multiplication is associative
and distributive.

A field is a commutative ring
with multiplicative identity
and inverses for all except 0.
Question
 Are there rings which are not
fields?

Examples:
• Z - no multiplicative
inverses
• M2(R) - not a commutative
ring
Complex Field

The group of complex
numbers (C, +) are
isomorphic to (R2, +).
•
= {(a, b): a, b  R}
• Map: a+bi  (a, b)
• (C, +) is commutative
R2

Prove (C, +, ) is a field.

Multiplication on C is
commutative.
• (a, b)(c, d) = (ac-bd, ad+bc) =
(ca-db, da+cb) = (c, d)(a, b)


Multiplication is defined on C
(or R2).
• (a, b)(c, d) = (ac-bd, ad+bc)

The multiplicative identity is
(1, 0).
Every non-zero element has
an inverse.
• (a, b)-1 = (a/a2+b2, -b/a2+b2)
Vector Space

A vector space combines a
group and a field.

• v, u  V; f, g  F
• (V, +) a commutative group
• (F, +, ) a field


Elements in V are vectors
• matrices, polynomials,
functions

Elements in F are scalars
• reals, complex numbers
Scalar multiplication
S1
provides the combination.



Closure: fv  V
Identity: 1v = v
Associative: f(gv) = (fg)v
Distributive:
• f(v+u) = fv + fu
• (f+g)v = fv + gv
Cartesian Vector

x2
A real Cartesian
vector is
S1
made from a Cartesian
product of the real numbers.
• EN = {(x1, …, xN): xi  R}
• Addition by component
• Multiplication on each
component
(x1, x2)
x1

This specific type of a vector
is what we think of as having
a “magnitude and direction”.
Algebra


An algebra is a linear vector
space with vector
multiplication.
Algebra definitions: v,w,x  V
• Closure: v□w  V
• Bilinearity:
(v+w)□x = v□x + w□x
x□(v+w) = x□v + x□w

Some algebras have
additional properties.
• Associative: v □(w□x) =
(v□w)□x
• Identity:  1  V, 1v = v1 = v,
vV
• Inverse:  v-1  V, v-1v = vv-1
= 1,  v  V
• Commutative: v□w = w□v
• Anticommutative: v□w=-w□v
Quaternions
 1 i
j
k
1 1 i
j
k
i i -1 k - j
j j - k -1 i
k k
j - i -1

Define a group
1 with addition
S
on R4.
• Q = {(a1, a2, a3, a4): ai  R}
• Commutative group

Define multiplication of 1, i, j,
k by the table at left.

Multiplication is not
commutative.
The quaternions are not
isomorphic to the cyclic 4-group.
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