For any two vector spaces [math]V[\/math] and [math]W[\/math], the tensor product [math]V\\bigotimes W[\/math] is the space of bilinear functions on [math]V\\times W[\/math] .<\/p>\n
If [math]V[\/math] and [math]W[\/math] are inner product spaces then for any [math]v\\in V[\/math] and [math]w\\in W[\/math] we can define the pure tensor [math]v\\otimes w[\/math] in [math]V\\bigotimes W[\/math] by [math] v\\otimes w (v*,w*)=(v\\cdot v* )(w\\cdot w*)[\/math] also often written by physicists as [math]\\langle v|v*\\rangle \\langle w|w*\\rangle[\/math]. But there are also elements of [math]V\\bigotimes W[\/math] that are not of the pure tensor form.<\/p>\n
For example if [math]v_1\\otimes w_1+v_2\\otimes w_2[\/math] could be written in the form [math](a_1v_1+a_2v_2)\\otimes(b_1w_1+b_2w_2)[\/math], then for all [math]v\\in V[\/math] and [math]w\\in W[\/math], we’d need [math]\\begin{align}&(a_1v_1+a_2v_2)\\otimes(b_1w_1+b_2w_2)(v,w)\\\\ &=(a_1v_1\\cdot v)(b_1w_1\\cdot w)+(a_1v_1\\cdot v)(b_2w_2\\cdot w)\\\\&+(a_2v_2\\cdot v)(b_1w_1\\cdot w)+(a_2v_2\\cdot v)(b_2w_2\\cdot w)\\\\&=(v_1\\cdot v)(w_1\\cdot w)+(v_2\\cdot v)(w_2\\cdot w)\\end{align}[\/math] .<\/p>\n
But this is only true if [math]a_1b_1=a_2b_2=1[\/math] and [math]a_1b_2=a_2b_1=0[\/math], but if one of [math]a_1[\/math] or [math]b_2[\/math] is zero then one of [math]a_1b_1[\/math] or [math]a_2b_2[\/math] must be also.<\/p>\n
Source: (405) What is a tensor product in simple words? – Quora<\/a><\/em><\/p>\n\n\n <\/p>\n","protected":false},"excerpt":{"rendered":" For any two vector spaces [math]V[\/math] and [math]W[\/math], the tensor product [math]V\\bigotimes W[\/math] is the space of bilinear functions on [math]V\\times W[\/math] . If [math]V[\/math] and [math]W[\/math] are inner product spaces then for any [math]v\\in V[\/math] and [math]w\\in W[\/math] we can define the pure tensor [math]v\\otimes w[\/math] in [math]V\\bigotimes W[\/math] by [math] v\\otimes w (v*,w*)=(v\\cdot … Continue reading (405) What is a tensor product in simple words? – Quora<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"topics":[],"class_list":["post-458","post","type-post","status-publish","format-standard","hentry","category-all"],"_links":{"self":[{"href":"https:\/\/qpr.ca\/blogs\/physics\/wp-json\/wp\/v2\/posts\/458","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/qpr.ca\/blogs\/physics\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/qpr.ca\/blogs\/physics\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/qpr.ca\/blogs\/physics\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/qpr.ca\/blogs\/physics\/wp-json\/wp\/v2\/comments?post=458"}],"version-history":[{"count":2,"href":"https:\/\/qpr.ca\/blogs\/physics\/wp-json\/wp\/v2\/posts\/458\/revisions"}],"predecessor-version":[{"id":461,"href":"https:\/\/qpr.ca\/blogs\/physics\/wp-json\/wp\/v2\/posts\/458\/revisions\/461"}],"wp:attachment":[{"href":"https:\/\/qpr.ca\/blogs\/physics\/wp-json\/wp\/v2\/media?parent=458"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/qpr.ca\/blogs\/physics\/wp-json\/wp\/v2\/categories?post=458"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/qpr.ca\/blogs\/physics\/wp-json\/wp\/v2\/tags?post=458"},{"taxonomy":"topics","embeddable":true,"href":"https:\/\/qpr.ca\/blogs\/physics\/wp-json\/wp\/v2\/topics?post=458"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}