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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.

Ҳоло шумо Ҳеҷ ҳисобкунак инро ҳал намекунад, лекин қисмҳои он ҳисобшавандаанд. Яке аз инҳоро дар поён санҷед ё худи худро ворид кунед.

Кор кардан

Дар ҳисоби ройгон ба ҳар як дарс қайдҳо илова карда мешаванд, қайди он, ки шумо чӣ кор кардаед, масъалаҳои ҳалшуда дар як ҷо ва муаллиме, ки шумо метавонед дар бораи ин саҳифа пурсед. Математика барои ҳама кушода аст, хоҳ ворид шуда бошад ё на.

Бақайдгирӣ Ворид шудан

Символы, используемые здесь

Барои муайянкунии пурраи маъно, тасвир ва маънои ҳар як ҳарф дар он, ба ҳар як аломат пахш кунед.

Саволҳои маъмул

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.

Қисми ин саҳифа аз Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Дар ин ҷо барои мо чизе нест, ки мо ба он бирасем.

& Иловаи забон 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