{"id":157,"date":"2023-07-20T15:26:55","date_gmt":"2023-07-20T14:26:55","guid":{"rendered":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/?p=157"},"modified":"2023-07-21T06:16:34","modified_gmt":"2023-07-21T05:16:34","slug":"change-of-basis-of-a-linear-map","status":"publish","type":"post","link":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/2023\/07\/20\/change-of-basis-of-a-linear-map\/","title":{"rendered":"Change of basis of a linear map"},"content":{"rendered":"<h3>1. Coordinate maps<\/h3>\n<p>consider a random vector <strong><em>a<\/em><\/strong>. If the values of every entry are explicitly given, one could naturally identify it as a vector under the orthonormal basis. However, it could also been understood as the <em>coordinate vector<\/em>, which represents a vector under another basis by a <em>coordinate map<\/em>:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1375\" height=\"46\" class=\"alignnone size-full wp-image-158\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/07\/equation1-1.png\" alt=\"\" \/><\/p>\n<h3>2. Change of basis of a linear map<\/h3>\n<p>Now we consider a linear map (represented by a matrix) A. It could be interpreted \u00a0as:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"230\" height=\"21\" class=\"alignnone size-full wp-image-162\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/07\/CodeCogsEqn-2.gif\" alt=\"\" \/><\/p>\n<p>What if we want to change the basis of the objective vector space of <em>f<\/em> ? The following relation is obvious, if we want to find a matrix <em>A&#8217;<\/em> representing the map after change of basis:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"462\" height=\"21\" class=\"alignnone size-full wp-image-165\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/07\/CodeCogsEqn-3.gif\" alt=\"\" \/><\/p>\n<p>Where <em>P<\/em> is defined as a matrix that perform this basis change.<\/p>\n<p><strong>[Literature: Andre Lukas, Lecture note on Vectors and Matrices, University of Oxford]<\/strong><\/p>\n<h3>3. The invariant map<\/h3>\n<p>Suppose we have a map that is invariant under any basis change, that is:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"240\" height=\"21\" class=\"alignnone size-full wp-image-167\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/07\/CodeCogsEqn-4.gif\" alt=\"\" \/><\/p>\n<p>In other words, we would like to find an operator that commutes with any other operator on the same vector space <em>V<\/em>. Suppose now we have a vector <strong><em>x<\/em><\/strong> in <em>V<\/em>. We would like to find a non-trivial linear functional on <em><strong>x<\/strong><\/em>, so that we can define a linear map:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"343\" height=\"19\" class=\"alignnone size-full wp-image-169\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/07\/CodeCogsEqn-5.gif\" alt=\"\" \/><\/p>\n<p>This is possible, for a functional in a <em>n <\/em>dimensional space can be express in to an <em>(1 x n)<\/em> matrix, so the left hand side can be expressed as <em>(nx1)(1xn)(nx1)<\/em> corresponding to <strong><em>v<\/em><\/strong>, <em>f<\/em>, and <strong><em>x<\/em> <\/strong>respectively, and the former 2 combined and form a matrix. ( <strong><em>My reference mentioned &#8221; Axiom of choice&#8221; with respect to finding non-trivial functional, and yet I have not understood it perfectly<\/em><\/strong>)<\/p>\n<p>Then, the commutation relation implies:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"356\" height=\"19\" class=\"alignnone size-full wp-image-170\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/07\/CodeCogsEqn-6.gif\" alt=\"\" \/><\/p>\n<p>Note that <em>f(T<strong>x<\/strong>)<\/em>, according to our pervious argument, should exist and independent on <strong><em>v<\/em><\/strong>. Thus, what we are doing consequently is that we have constructed linear maps <em>P<\/em> according to our need (that is, the vector <strong><em>v<\/em><\/strong>). In other words, we can assign every vector <strong><em>v<\/em><\/strong> in <em>V<\/em> a matrix <em>P<\/em> and they have to satisfy (6).<\/p>\n<p>Then this inplies:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"234\" height=\"44\" class=\"alignnone size-full wp-image-173\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/07\/CodeCogsEqn-7.gif\" alt=\"\" \/><\/p>\n<p><em>i.e.<\/em> <strong><em>T is the scalar multiple of \u00a0the identity<\/em><\/strong>.<\/p>\n<p><strong>[Literature: Robert Isreal, https:\/\/math.stackexchange.com\/questions\/27808\/a-linear-operator-commuting-with-all-such-operators-is-a-scalar-multiple-of-the]<\/strong><\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>1. Coordinate maps consider a random vector a. If the values of every entry are explicitly given, one could naturally identify it as a vector under the orthonormal basis. However, it could also been understood as the coordinate vector, which represents a vector under another basis by a coordinate map: 2. Change of basis of [&hellip;]<\/p>\n","protected":false},"author":1741,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[19],"tags":[],"class_list":["post-157","post","type-post","status-publish","format-standard","hentry","category-linear-algebra"],"_links":{"self":[{"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/posts\/157","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/users\/1741"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/comments?post=157"}],"version-history":[{"count":8,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/posts\/157\/revisions"}],"predecessor-version":[{"id":175,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/posts\/157\/revisions\/175"}],"wp:attachment":[{"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/media?parent=157"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/categories?post=157"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/tags?post=157"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}