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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, and W a subset of V.

Subspaces

Just as groups have subgroups and rings have subrings, vector spaces also have substructures. Let \(V\) be a vector space over a field \(F\), 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 \(W\).

Example

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 \[\begin{aligned}\alpha (x_1, 2 x_1 + x_2, x_1 - x_2) & = (\alpha x_1, \alpha(2 x_1 + x_2), \alpha( x_1 - x_2)) \\ & = (\alpha x_1, 2(\alpha x_1) + \alpha x_2, \alpha x_1 -\alpha x_2)\end{aligned}\], \(W\) 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 \[\begin{aligned}\end{aligned}\].

Example

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 \[\begin{aligned}\end{aligned}\] 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 \(v_1, v_2, \ldots, v_n\).

Condensed: the full section is in Judson, Abstract Algebra: Theory and Applications.

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What is a group, in plain words?

A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group. That is where the idea came from.

What is the difference between a ring and a field?

A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.

Nalaina avy amin'ny Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Nofafana sy nanazavana indray eto; ny hadisoana dia antsika.

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