## 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 \[V\]

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.