Subspaces

Suppose we have a vector space  
\[V\]
, with a subset  
\[S\]
  of vectors in  
\[V\]
.
Then  
\[S\]
  is said to be a subspace of  
\[V\]
  if
For all scalars  
\[\alpha,\:\beta\]
  and vectors  
\[\mathbf{v_1},\:\mathbf{v_2} \in S, \alpha \mathbf{v_1} + \beta \mathbf{v_2} \in S\]
.
Notice that this implies that  
\[S\]
  must contain the zero vector, by taking  
\[\alpha = \beta =0\]
.
The set of real numbers  
\[\mathbb{R}\]
  is a subspace of the vector space of complex numbers. To see this notice that  
\[0 \in \mathbf{R}\]
  and that if  
\[\alpha, \: \beta\]
  are scalars, hence real numbers, and  
\[r_1, r_2 \in \mathbb{R}\]
  then  
\[\alpha r_1 + \beta r_2 \in \mathbb{R}\]
.
Every k plane through the origin is a subspace of  
\[\mathbb{R}^n\]
  if  
\[k \leq n\]
.
The sets of diagonal matrices, lower and upper triangular matrices, square matrices for each value of  
\[n\]
, the set  
\[P_n\]
  of polynomials of degree at most  
\[n\]
  are all subspaces of the obvious vector spaces.