Vector spaces and subspacesWe have alr e a dy encountered vectors inRn. Now, we discuss the gen e r a l concept ofvectors.In place of the spaceRn, we think of general vector spaces.Definition 1. A vector space is a nonempty set V of elements, called vectors, whichmay be added and scaled (mu ltiplied with real numbers) .The two operati ons of addition and scalar multiplicatio n must sa tisfy th e followingaxioms for allu, v , w in V , and all scalars c, d.(a) u + v is in V(b) u + v = v + u(c) (u + v) + w = u + (v + w)(d) there is a vector (called the zero ve ctor) 0 in V such that u + 0 = u for all u in V(e) there is a vector −u such tha t u + (−u) = 0(f) cu is in V(g) c(u + v) = cu + cv(h) (c + d)u = cu + du(i) (cd)u = c(du)(j) 1u = utl;drA vector space is a collection of vectors which can be added and scaled;subject to the usual rules you would hope for.namely: associativity, commutativity, distributiv ityArmin [email protected] 2. Convince yourself that M2×2=na bc d: a, b, c, d in Rois a vector space.Solution. In th is context, the zero ve c tor is 0 =Example 3. Let Pnbe the set of all polynomia ls of degree a t most n > 0. Is Pnavector space?Solution.Armin [email protected] 4. Let V be the set of all polynomials of degree exactly 3. Is V a vector spac e ?Solution.Example 5. Let V be the set of all functions f : R → R. Is V a vector space?Solution.Subspac esDefinition 6. A subset W of a vector space V is a subspace if W is itself a vector space .Armin [email protected] the rules li ke as sociati vity, commutativity and distrib utivity still hold, we only needto check the following:W ⊆ V is a subspace of V if• W contains the zero vector 0,• W is closed under addition, (i.e. if u, v ∈ W then u + v ∈ W )• W is closed under scaling. (i.e. if u ∈ W an d c ∈ R then cu ∈ W )Example 7. Is W = spann11oa subspace of R2?Solution.Example 8. Is W =(a0b: a, b in R)a subs pace of R3?Solution.Armin [email protected] 9. Is W =n00oa subs pace of R2?Solution.Example 10. Is W =nxx + 1: x in Roa subspace of R2?Solution.Spans of vectors are subspacesReview. The span of vectors v1, v2,, vmis t he set of a ll their linear combin ations.We denote it byspan{v1, v2,, vm}.In other words,span{v1, v2,, vm} is the set of all vectors of th e formc1v1+ c2v2++ cmvm,wh ere c1, c2,, cmare scalars.Armin [email protected] 11. If v1,,vmare in a v e c tor space V , then span{v1,, vm} is a subspaceof V .Example 12. Is W =na + 3b2a − b: a, b in Roa subs pace of R2?Solution.Example 13. Is W =n−a 2ba + b 3a: a, b in Roa subspace of M2×2, the space of 2 × 2matrices?Solution.Armin [email protected] problemsExample 14. Are the following vector spaces?(a) W1=(abc: a + 3b = 0, 2a − c = 1)(b) W2=a + c−2bb + 3cc: a, b, c in R(c) W2=nab: ab > 0oArmin
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