A linear vector space over a scalar set (we shall typically consider sets or ) is a set of objects (called vectors) , , , with two operations:
These must satisfy the following conditions
and associative
and associative,
Note that we have not defined subtraction; it is derived operation, and is defined through the addition of an inverse element.
Example 2.1:
The space of vectors is a vector space over the set .
Example 2.2:
The space of two-dimensional complex spinors
, is a vector space.
Note: If we look at the space of up and down spins, we must require that the length of the vectors (the probability), , is 1. This is not a vector space, since
which is not necessarily equal to 1.
Example 2.3:
The space of all square integrable (i.e., all functions with ) complex functions of a real variable, is a vector space, for .
The space defined above is of crucial importance in Quantum Mechanics. These wave functions are normalisable (i.e., we can define one with total probability 1).
Show that the zero vector is unique, and that for each there is only one inverse .