## 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.