Vector space

Now that we have covered all of the abstract concepts we need to understand, we can give a formal definition of a vector space, before looking at the implications of these in the following chapters.

A vector space is defined as having the following mathematical objects:

1. An Abelian group ⟨V,+⟩ with an identity element e. We call members of the set V vectors. We define the identity element to be the zero vector, and we denote this by 0. The operation + is called vector addition.
2. A field {F, +, ⋅}. We say that V is a vector space over the field F, and we call the members of F scalars.

   The Zero Vector Is Not Denoted by |0⟩

   It is important to note that we denote the zero vector with a bold zero – 0 – and it is totally different from the vector |0⟩ we defined earlier in the book. This is the convention in quantum computing.

We can define an additional operation as scalar multiplication, which is an operation...