It gives you a simple recipe to check whether a subset of a vector space is a supspace.
If W is a subset of a vector space V and if W is itself a vector space under the inherited operations of addition and scalar multiplication from V, then W is called a subspace.
Rather the fact that "nonempty and closed under multiplication and addition" are (necessary and) sufficient conditions for a subset to be a subspace should be seen as a simple theorem, or a criterion to see when a subset of a vector space is in fact a subspace. Subspaces - Examples with Solutions Definiton of Subspaces. So this way there is no real difference, and one should better introduce and define the notion of subspace per "vectorspace that is contained (the way I describe above) in a vector space" instead of "subset with operations that have some magical other properties". So subspace implies all of these things, and all of these things imply a subspace. If I have a subset of Rn, so some subset of vectors of Rn, that contains the 0 vector, and its closed under multiplication and addition, then I have a subspace. Actually, there is a reason why a subspace is called a subspace: It is also a vector space and it happens to be (as a set) a subset of a given space and the addition of vectors and multiplicataion by scalars are "the same", or "inherited" from that other space. You dont somehow end up with a vector thats outside of your set. In general, all ten vector space axioms must be veried to show that a set W with addition and scalar multiplication forms a vector space. For any vector AW and a scalar c, the scalar. You should not want to distinguish by noting that there are different criteria. A subset W of a vector space V is called a subspace of V if W is itself a vector space under the addition and scalar multiplication dened on V. Subspaces in General Vector Spaces The zero vector in V is in W. The number of axioms is subject to taste and debate (for me there is just one: A vector space is an abelian group on which a field acts). We say that a subset U of a vector space V is a subspace of V if U is a vector space under the inherited addition and scalar multiplication operations of V.
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