\n

\n Question is in the picture.\n <\/p>\n

\n Algebra 2.\n <\/p>\n

\n MATH 236 in mcgill\n <\/p>\n

Exercise 7. Let V be a \ufb01nite\u2014dimensional vector space over IF. The vector space V* :\u00a3(V, F) is called the dual of V. An element of V* is thus a linear map f : V \u2014&gt; IF, alsoknown as a functional. (i) Let {U1,. .. ,vn} be a basis of V. Show that, for each i E {1, . . . ,n}, there exists aunique \u201903\u201c E V* satisfying1 if 2′ : j vi(vj)6i'{0 ifz\u20197\u00e9j’ Show that {01\u201c, . . . , 11;} is a basis of V*. Deduce that V and V* are isomorphic.(ii) For each 1: E V, let evv : V* \u2014&gt; IF denote the evaluation at 1); that is, evu(f) : f (v)for each f E V*. Check that evv E V**, the dual of V*. Show that the evaluationmap ev : V \u2014&gt; V** is an isomorphism. (A brief comment. The isomorphism of V and V** is canonical, in the sense that it doesnot depend on a basis. The isomorphism of V and V*, on the other hand, is not canonical\u2014it requires the choice of a basis. The notion of duality, and the canonical\/non\u2014canonicalisomorphisms exhibited above, appear in relation to several mathematical structures.)\n<\/p><\/div>\n","protected":false},"excerpt":{"rendered":"

Question is in the picture. Algebra 2. MATH 236 in mcgill Exercise 7. Let V be a \ufb01nite\u2014dimensional vector space over IF. The vector space V* :\u00a3(V, F) is called the dual of V. An element of V* is thus a linear map f : V \u2014&gt; IF, alsoknown as a functional. (i) Let {U1,. […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_joinchat":[]},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/qualityassignments.net\/wp-json\/wp\/v2\/posts\/228473"}],"collection":[{"href":"https:\/\/qualityassignments.net\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/qualityassignments.net\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/qualityassignments.net\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/qualityassignments.net\/wp-json\/wp\/v2\/comments?post=228473"}],"version-history":[{"count":0,"href":"https:\/\/qualityassignments.net\/wp-json\/wp\/v2\/posts\/228473\/revisions"}],"wp:attachment":[{"href":"https:\/\/qualityassignments.net\/wp-json\/wp\/v2\/media?parent=228473"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/qualityassignments.net\/wp-json\/wp\/v2\/categories?post=228473"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/qualityassignments.net\/wp-json\/wp\/v2\/tags?post=228473"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}