Just as groups have subgroups and rings have subrings, vector spaces also have substructures. Let \(V\) be a vector space over a field \(F\), and \(W\) a subset of \(V\). 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 \(\alpha \in F\), it will always be the case that \(u + v\) and \(\alpha v\) are also in \(W\).

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Example20.6

Let \(W\) be the subspace of \({\mathbb R}^3\) defined by \(W = \{ (x_1, 2 x_1 + x_2, x_1 - x_2) : x_1, x_2 \in {\mathbb R} \}\). We claim that \(W\) is a subspace of \({\mathbb R}^3\). Since
\begin{align*}
\alpha (x_1, 2 x_1 + x_2, x_1 - x_2) & = (\alpha x_1, \alpha(2 x_1 + x_2), \alpha( x_1 - x_2))\\
& = (\alpha x_1, 2(\alpha x_1) + \alpha x_2, \alpha x_1 -\alpha x_2),
\end{align*}
\(W\) 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 \(W\). Then
\begin{equation*}u + v = (x_1 + y_1, 2( x_1 + y_1) +( x_2 + y_2), (x_1 + y_1) - (x_2+ y_2)).\end{equation*}

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Example20.7

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 \(p(x) + q(x)\). Also, \(\alpha p(x) \in W\) for \(\alpha \in F\) and \(p(x) \in W\).

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 \(F\). Any vector \(w\) in \(V\) of the form
\begin{equation*}w = \sum_{i=1}^n \alpha_i v_i = \alpha_1 v_1 + \alpha_2 v_2 + \cdots + \alpha_n v_n\end{equation*}
is called a *linear combination* of the vectors \(v_1, v_2, \ldots, v_n\). The *spanning set* of vectors \(v_1, v_2, \ldots, v_n\) is the set of vectors obtained from all possible linear combinations of \(v_1, v_2, \ldots, v_n\). If \(W\) is the spanning set of \(v_1, v_2, \ldots, v_n\), then we say that \(W\) is *spanned* by \(v_1, v_2, \ldots, v_n\).

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Proposition20.8

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

Let \(u\) and \(v\) be in \(S\). We can write both of these vectors as linear combinations of the \(v_i\)'s:
\begin{align*}
u & = \alpha_1 v_1 + \alpha_2 v_2 + \cdots + \alpha_n v_n\\
v & = \beta_1 v_1 + \beta_2 v_2 + \cdots + \beta_n v_n.
\end{align*}
Then
\begin{equation*}u + v =( \alpha_1 + \beta_1) v_1 + (\alpha_2+ \beta_2) v_2 + \cdots + (\alpha_n + \beta_n) v_n \end{equation*}
is a linear combination of the \(v_i\)'s. For \(\alpha \in F\),
\begin{equation*}\alpha u = (\alpha \alpha_1) v_1 + ( \alpha \alpha_2) v_2 + \cdots + (\alpha \alpha_n ) v_n \end{equation*}
is in the span of \(S\).