14/8/12: Linear Vector Space

What is a vector ?
A vector is much more than the high-school definition of "something with a magnitude and direction" (so is Usain Bolt a vector???)
A vector is defined in general as an element of a vector space

What is a vector space ?
A set of certain abstract objects represented by the symbol |.>, e.g., |a> (using Dirac notation) for which addition of a pair of elements and multiplication by a scalar (a real or complex number) are defined.