\[V\]
, with a subset \[S\]
of vectors in \[V\]
.Then
\[S\]
is said to be a subspace of \[V\]
ifFor 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.