Vector Spaces

Prerequisites:

Groups

A Vector Space has four ingredients --

A set, V, whose elements are called vectors.
A particular vector, O, called the zero vector.
An operation, called vector addition. The result of applying this operation to two vectors, u and v, is another vector, written u + v.
An operation, called scalar multiplication, which takes a real number c and a vector v and produces a new vector, written cv.

These operations have the following properties, called the vector space properties --

V with vector addition as its operation and the zero vector as its identity element is an Abelian group.
If u and v are any vectors and a and b are any real numbers then
1u = u.
0u = O.
a(bu) = (ab)u.
(a + b)u = (au + bu).
a(u + v) = (au) + (av).

Recall that in a group every element has an inverse. In a vector space, the inverse of a vector v is the vector (-1)v as shown by the short calculation below.


v + (-1)v = 1v + (-1)v
          = (1 + -1)v
          = 0v
          = O.

From the short list of vector space properties we can deduce many other important properties of vector spaces.

Theorem: If v is a vector, a is a real number and

av = O

then either a = 0 or v = O.

Proof

if a is not zero then we can multiply both sides of the equation

 av = O

by (1/a) to get


(1/a)(av) = O 
((1/a)a)v = O
       1v = O
        v = O

Definition:

If u and v are two vectors we define u - v by

u - v = u + (-1)v

Theorem:

If u and v are vectors then

u + (v - u) = v

Proof:


u + (v - u) = u + (v + (-1)u)
            = 1u + ((-1)u + v)
            =  (1u + (-1)u) + v
            = (1 + (-1))u + v
            = 0u + v
            = O + v
            = v

Examples:

A quadratic function is any function that can be described by a formula of the form
f(x) = ax² + bx + c
Notice one or more of the coefficients a, b, or c may be zero. In fact, they can all be zero.
Let V be the set of all quadratic functions and define two operations on V in the usual way by

If f and g are two quadratic functions then the function f + g is given by
(f + g)(x) = f(x) + g(x)
and if c is a real number then the function cf is given by
(cf)(x) = c f(x)

Is V with these two operations a vector space? If so, what is the zero vector? answer
Let F be the set of all functions, f(x), that are defined for every real number and are zero at 1 -- that is, such that f(1) = 0. Define f + g and cf in the usual way.
Is F with these two operations a vector space? answer
Let W be the set of all functions, f(x), that are defined for every real number and such that f(1) = 2. Define f + g and cf in the usual way.
Is W with these two operations a vector space? answer