**Definition 1 (Vector Space)**. A vector space over a field F is a quadruple (V, +, ., F )) satisfying the following axioms for all α, β ∈ F and x, y, z ∈ V :

1. (V, +) is a commutative group, that is,

(a) ’+’ is map from V x V to V. [Closure]

(b) (x + y) + z = x + (y + z). [Associative]

(c) there exists an element 0 of V such that x + 0 = 0 + x = x. [Existence of 0]

(d) for each x in V there exists an element −x in V such that x + (−x) = (−x) + x = 0.

[Existence of Negative]

(e) x + y = y + x. [Commutative]

2. ’.’ is a map from F xV to V. [Closure wrt . ]

3. α.(β.x) = (α.β).x.

4. 1.x=x

5. (α + β).x = (α.x) + (β.x). [Distributivity]

6. α.(x + y) = (α.x) + (α.y). [Distributivity]**Remark 1**. The elements of V are called vectors and the elements of F are called scalars. F is called the **base field or ground field** of the vector space. ’0’ in axiom 1(c) is called the null vector or the zero vector. It is to noted that, the bold faced Roman letters are treated as vectors whereas the lower case Greek letters are treated as scalars throughout the course.

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