Definition of Vector Spaces

VECTOR SPACES

Definition:

A vector space (or linear space) V over a field "F" consists of a set on which two operations (called addition & scalar multiplication) are defined so that for each pair of elements x, y ∈ V there is a unique element x+y ∈ V and for each element "a" in F and each element x in V there is a unique element ax in V, such that the following conditions hold.

1. For all x, y ∈ V, x+y=y+x (Commutative law)

2. For all x, y, z ∈ V, (x + y) + z = x + (y + z). (Associative law)

3. There exists an element in V denoted by 0 such that x+0= x for each x in V.

4. For each element x in V there exists an element y in V such that x+y=0.

5. For each element x in V, 1.x = x.

6. For each pair of elements a, b ∈ F and each element x in V (a b) x = a (bx).

7. For each element "a" in F and each pairs of elements x, y in V, a(x + y) = ax + ay.

8. For each pair of elements a, b in F and each element x in V, (a + b)x = ax + bx.

The elements of the field F are called scalars and the elements of the vector space V are called vectors.

An object of the form (a₁, a₂, a₃….. a_n) where the entries a₁, a₂, a₃….. a_n an are elements of a field F, is called an n−tuples with entries from F. The elements a₁, a₂, a₃….. a_n an are called the entries or components of the n−tuple. Two n−tuples (a₁, a₂, a₃….. a_n) and (b₁, b₂, ... b_n) with entries from a field F are called equal if a_i= b_i for i = 1, 2 ... n.

Vectors in F may be written as column vectors, 

An _m×n matrix with entries from a field F is a rectangular array of the form

where each entry a_ij (1 ≤  i ≤ m, 1 ≤  j ≤ n) is an element of F. We call the entries a_ij with i=j is the diagonal entries of the matrix. The entries a_i1, a_i2, a_i3, ...a_in compose the i^th of the matrix, and the entries a_1j, a_2j, a_3j, ...a_mj compose the j^th column of the matrix.

The m×n matrix in which each entry equals zero is called the zero matrix and it is denoted by O. Here we denote matrices by capital letters and we denote the entry of a matrix A that lies in row i and column j by A_ij. If the number of rows and columns of a matrix are equal, the matrix is called a square matrix. Two m×n matrices A and B are called equal if all their corresponding entries are equal, if A_ij=B_ij for 1 ≤ i ≥ m and 1 ≤ i ≥ m.