What is vector space?

Answer 1

A vector space is an algebraic structure with two binary operations, + and *, that satisfy the following axioms:

1) Associativity: a + ( b + c ) = ( a + b ) + c

2) Commutativity: a + b = b + a

3) Addition identity: a + 0 = a

4) Additive inverse: a + a-1 = 0

5) Scalar distributivity with respect to vector addition: c*(a + b) = c*a + c*b

6) Scalar distributivity with respect to field addition: (c + d)*a = c*a + d*a

7) Compatibility: c*(d*a) = (c*d)*a

8) Scalar multiplicative identity: 1*a = a

For the above axioms a, b, 0, Є Vand 1, c, d Є F where V is a set of vectors over the field F. Most of the time F = R or C, which are the sets of real and complex numbers, respectively.