Vector Spaces

vector space

Definition

A real vector space is a set \(\displaystyle V\) along with two binary operations \(\displaystyle +,\cdot\):

such that the following axioms hold.

Commutativity: \(\displaystyle u+v=v+u\) for all \(\displaystyle u,v\in V\)
Associativity: \(\displaystyle ( u+v) +w=u+( v+w)\) for all \(\displaystyle u,v,w\in V\)
Additive identity: there exists an element in \(\displaystyle V\) denoted as \(\displaystyle 0\) such that \(\displaystyle 0+v=v\) for all \(\displaystyle v\in V\)
Additive inverse: for every \(\displaystyle v\in V\), there exists an element denoted as \(\displaystyle -v\) such that \(\displaystyle v+( -v) =0\)

Associativity: \(\displaystyle ( ab) v=a( bv)\) for all \(\displaystyle a,b\in \mathbb{R}\) and \(\displaystyle v\in V\)
Multiplicative identity: \(\displaystyle 1v=v\) for all \(\displaystyle v\in V\)
Distributivity
- \(\displaystyle ( a+b) v=av+bv\) for all \(\displaystyle a,b\in \mathbb{R}\) and \(\displaystyle v\in V\)
- \(\displaystyle a( u+v) =au+av\) for all \(\displaystyle u,v\in V\) and \(\displaystyle a\in \mathbb{R}\)

The set \(\displaystyle \mathbb{R}^{2}\) with the usual rules of addition and scalar multiplication is a vector space.
As an extension, the set \(\displaystyle \mathbb{R}^{n}\) with the usual rules of addition and scalar multiplication is a vector space.
The set of all polynomials of degree at most \(\displaystyle n\) with real coefficients is a vector space with the usual rules of polynomial addition and scalar multiplication. This is sometimes denoted as \(\displaystyle P_{n}(\mathbb{R})\).
The set of all real \(\displaystyle m\times n\) matrices is a vector space. This is sometimes denoted as \(\displaystyle M_{m\times n}(\mathbb{R})\).

Some of the familiar rules of algebra carry over:
- \(\displaystyle av=0\Longrightarrow a=0\) or \(\displaystyle v=0\) or both, where \(\displaystyle a\in \mathbb{R}\) and \(\displaystyle v\in V\)
- \(\displaystyle -v=( -1) v\), where \(\displaystyle v\in V\)
- \(\displaystyle u+v=0\Longrightarrow u=-v\), where \(\displaystyle u,v\in V\)
For a given vector space, the zero element is unique. The zero element plays a very important role in the theory of vector spaces.
The same symbol is used to denote both the scalar zero and the vector zero. The context will make this distinction clear.
The zero element for some of the vector spaces discussed above:
- For \(\displaystyle \mathbb{R}^{2}\), it is \(\displaystyle ( 0,0)\).
- For \(\displaystyle \mathbb{R}^{n}\), it is \(\displaystyle ( 0,\cdots ,0)\)
- For \(\displaystyle P_{2}(\mathbb{R})\) it is \(\displaystyle p( x) =0+0x+0x^{2}\), the zero polynomial.
- For \(\displaystyle M_{2\times 2}(\mathbb{R})\), it is the zero matrix of size \(\displaystyle 2\times 2\), \(\displaystyle \begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix}\).