Definition of Vector Space and its Theorem

Scalar addition and multiplication operations are used in diverse contexts in mathematics. Regardless of the context, however, these operations usually fulfill the same rules of arithmetic. So the general theory of mathematical systems involving scalar addition and multiplication will be applicable to various fields of mathematics. Mathematical systems with this form are called Vector Rooms or Linear Space. In this article we explain the definition of vector space and develop some general theories about vector space.

Vector Space

We can review Rn as a set of all n x 1 matrix with real number entries. Scalar addition and multiplication for vectors in Rn are the usual sum and scalar multiplication of the matrix. More generally, suppose Rm x Rn expresses the set of all matrix m x n with real number entries.
If A = (aij) and B = (bij), then the number of A + B is defined as the matrix C = (cij) which has the order m x n, where cij = aij + bij.
If given scalar α, then we can define αA as the matrix m x n where the ij entry is αaij.

Axiom of the Vector Space

Let V be the set where the scalar addition and multiplication operations are defined. By this we mean that for each pair of elements x and y in V, we can associate it with a single x + y element which is also at V, and with each element x in V and every scalar α, we can associate it with single αx element inside V.
The set V together with scalar addition and multiplication operations is said to form a vector space if the following axioms are fulfilled.

A1 x + y = y + x for every x and y in V.
A2 (x + y) + z = x + (y + z) for every x, y, z in V.
A3 There is an element 0 at V so x + 0 = x for every x ε V.
A4 For each x ε V there is a -x element in V so x + (-x) = 0.
A5 α (x + y) = αx + αy for each α scalar and every x and y in V.
A6 (α + β) x = αx + βx for each scalar α and β and every x ε V.
A7 (α β) x = α (β x) for each scalar α and β and every x ε V.
A8 x = x for each x ε V

Vector Space C | a, b |

Suppose C [a, b] expresses the set of all real-valued functions that are defined and continuous at the closing interval [a, b]. In this case the set should be a set of functions. So the vectors are functions in C [a, b]. The number of f + g of the two functions in C [a, b] is defined by

(f + g) (x) = f(x) + g(x)

For all x in [a, b]. The new function f + g is an element of C [a, b], because the sum of two continuous functions is continuous.

If f is a function in C [a, b] and α a real number, then α f is defined by

f) (x) = α f (x)

For all x in [a, b]. It is clear that αf is in C [a, b] because if a constant multiplied by a continuous function is always continuous.
So in C [a, b] we have defined operations — scalar addition and multiplication operations. To show that the first axiom is

f + g = g + f

fulfilled, we must show that

(f + g) (x) = (g + f) (x) for each x ϵ [a, b]

This equation is correct because

(f + g) (x) = f (x) + g (x) = g (x) + f (x) = (g + f) (x) for each x in [a, b]

Additional Properties of the Vector Room
From the explanation above can be summarized with a theorem which states three more basic properties for vector space, which will be explained by the following theorem,

Theorem 1.
If V is a vector space and x is any element of V, then
(i)        0x = 0
(ii)        x + y = 0 results in y = -x (that is, the inverse of x is singular)
(iii)       (-1) x = -x

Proof:
Based on the A6 and A8 axioms then
x = 1x = (1 + 0) x = 1x + 0x = x + 0x
so
- x + x = - x + (x + 0x) = (-x + x) + 0x ...... (A2)
0 = 0 + 0x = 0x ........................................ (A1, A3 and A4)
To prove (ii), suppose that x + y = 0, then
- x = - x + 0 = - x + (x + y)
Therefore,
- x = (-x + x) + y = 0 + y = y ...................... (A1, A2, A3 and A4)
Finally, to prove (iii), note that
0 = 0x = (1 + (-1)) x = 1x + (-1) x ................ [(i) and A6]
So
X + (-1) x = 0
And from section (ii) is obtained

(-) x = -x