maths.free › Abstract Algebra › Vector Spaces › Subspaces

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.

Sasa Hakuna kifaa cha kupigia hesabu kinachoweza kufanya hesabu, lakini kifaa hicho kinaweza kutumiwa katika hesabu.

Keep your own working

Akaunti la bure laongezea mambo makuu juu ya kila somo, rekodi ya yale ambayo umemaliza, matatizo yako ya kutatua katika sehemu moja, na mtunzaji unayeweza kuuliza juu ya ukurasa huu.

Jisajili Login

Ishara zinazotumiwa hapa

Vaa alama yoyote ya ufafanuzi kamili, picha, na maana ya kila herufi.

Maswali ambayo watu huuliza

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.

Sehemu za ukurasa huu zimebadilishwa kutoka kwa Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Tumekamatwa na kukosolewa tena hapa; makosa ni yetu.

Mengi zaidi katika Abstract Algebra

Groups Subgroups, cosets and Lagrange's theorem Cyclic groups and permutation groups Homomorphisms, normal subgroups and quotient groups Rings and fields Galois theory: why the quintic has no formula