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Vector Spaces

A vector space  
\[V\]
  is any set of objects with the properties
1.  
\[a \mathbf{V_1} + b \mathbf{V_2} \in V \]
  for all
\[V_1 , V_2 \in V\]
, for all scalars  
\[a,b\]

2.  
\[\mathbf{0} \in V\]

This definition is not restrictd to ordinary vectors in space. Any set of objects, when operated on by a linear operator, may be vonsidered a vector space.
Examples
The set of polynomials  
\[P_n\]
 with constant coefficients.
1.  
\[ \begin{equation} \begin{aligned} & a (a_0 + a_1 x+a_2 x^2 +...+ a_n x^n )+b (b_0 + b_1 x+b_2 x^2 +...+ b_n x^n ) \\ &=(aa_0 +bb_0) +(aa_1 +bb_1)x+(aa_2 +bb_2)x^2 +...+ (aa_n +bb_n) x^n \in P_n \end{aligned} \end{equation}\]

2.  
\[0 + 0x+0x^2 +..._ + 0x^n \in P_n\]

The set of functions returning real numbers is a vector space since sums and multiples of real numbersreal numbers and zero is a real number. The set of complex functions is a vector space for the same realson. The sets of all real numbers, complex numbers, the number zero by itself, can all be considered vector spaces. The set of all rational numbers is not a vector sapce, since if  
\[x \in \mathbb{R} , q \in \mathbb{Q} , xq \notin \mathbb{Q}\]
. The sets of natural numbers and integers are also not vector spaces.