## Section 20.2 Subspaces

Just as groups have subgroups and rings have subrings, vector spaces also have substructures. Let \(V\) be a vector space over a field \(F\text{,}\) and \(W\) a subset of \(V\text{.}\) Then \(W\) is a subspace of \(V\) if it is closed under vector addition and scalar multiplication; that is, if \(u, v \in W\) and \(\alpha \in F\text{,}\) it will always be the case that \(u + v\) and \(\alpha v\) are also in \(W\text{.}\)

###### Example 20.6.

Let \(W\) be the subspace of \({\mathbb R}^3\) defined by \(W = \{ (x_1, 2 x_1 + x_2, x_1 - x_2) : x_1, x_2 \in {\mathbb R} \}\text{.}\) We claim that \(W\) is a subspace of \({\mathbb R}^3\text{.}\) Since

\(W\) is closed under scalar multiplication. To show that \(W\) is closed under vector addition, let \(u = (x_1, 2 x_1 + x_2, x_1 - x_2)\) and \(v = (y_1, 2 y_1 + y_2, y_1 - y_2)\) be vectors in \(W\text{.}\) Then

###### Example 20.7.

Let \(W\) be the subset of polynomials of \(F[x]\) with no odd-power terms. If \(p(x)\) and \(q(x)\) have no odd-power terms, then neither will \(p(x) + q(x)\text{.}\) Also, \(\alpha p(x) \in W\) for \(\alpha \in F\) and \(p(x) \in W\text{.}\)

Let \(V\) be any vector space over a field \(F\) and suppose that \(v_1, v_2, \ldots, v_n\) are vectors in \(V\) and \(\alpha_1, \alpha_2, \ldots, \alpha_n\) are scalars in \(F\text{.}\) Any vector \(w\) in \(V\) of the form

is called a linear combination of the vectors \(v_1, v_2, \ldots, v_n\text{.}\) The spanning set of vectors \(v_1, v_2, \ldots, v_n\) is the set of vectors obtained from all possible linear combinations of \(v_1, v_2, \ldots, v_n\text{.}\) If \(W\) is the spanning set of \(v_1, v_2, \ldots, v_n\text{,}\) then we say that \(W\) is spanned by \(v_1, v_2, \ldots, v_n\text{.}\)

###### Proposition 20.8.

Let \(S= \{v_1, v_2, \ldots, v_n \}\) be vectors in a vector space \(V\text{.}\) Then the span of \(S\) is a subspace of \(V\text{.}\)

###### Proof.

Let \(u\) and \(v\) be in \(S\text{.}\) We can write both of these vectors as linear combinations of the \(v_i\)'s:

Then

is a linear combination of the \(v_i\)'s. For \(\alpha \in F\text{,}\)

is in the span of \(S\text{.}\)