{"id":18706,"date":"2016-11-15T21:39:23","date_gmt":"2016-11-15T21:39:23","guid":{"rendered":"http:\/\/www.oracletutoring.ca\/blog\/?p=18706"},"modified":"2016-11-15T21:39:23","modified_gmt":"2016-11-15T21:39:23","slug":"linear-algebra-projecting-a-vector-along-another-one-matrix-method","status":"publish","type":"post","link":"https:\/\/www.oracletutoring.ca\/blog\/linear-algebra-projecting-a-vector-along-another-one-matrix-method\/","title":{"rendered":"Linear algebra:  projecting a vector along another one, matrix method."},"content":{"rendered":"<h1>The tutor shows an example of a projection from one dimension into another.<\/h1>\n<p>Let&#8217;s imagine we have the vector <b>v<\/b><\/p>\n<p>3<br \/>\n2<br \/>\n0<\/p>\n<p>which we want to project in the direction of <b>w<\/b><\/p>\n<p>3<br \/>\n-1<br \/>\n0<\/p>\n<p>The projection matrix, P<sub>w<\/sub>, is given by <b>ww<sup>T<\/sup><\/b>\/<b>w<sup>T<\/sup>w<\/b>:<\/p>\n<p><img decoding=\"async\" src=\"\/..\/images\/proj_nov_15.png\" \/><\/p>\n<p>Now, the projection of <b>v<\/b> onto <b>w<\/b>, P<sub>w<\/sub><b>v<\/b>, is<\/p>\n<p><img decoding=\"async\" src=\"\/..\/images\/proj_nov_15_b.png\" \/><\/p>\n<p>Note that P<sup>2<\/sup><sub>w<\/sub><b>v<\/b> = P<sub>w<\/sub><b>v<\/b>:<\/p>\n<p><img decoding=\"async\" src=\"\/..\/images\/proj_nov_15_c.png\" \/><\/p>\n<p>Note, also, that P<sub>w<\/sub><b>v<\/b> has the same proportions as <b>w<\/b>.<\/p>\n<p>Source:<\/p>\n<p><a href=\"\nhttps:\/\/ocw.mit.edu\/courses\/mathematics\/18-06sc-linear-algebra-fall-2011\/least-squares-determinants-and-eigenvalues\/projections-onto-subspaces\/MIT18_06SCF11_Ses2.2sum.pdf\">mit.edu<\/a><\/p>\n<p>Jack of <a href=\"https:\/\/www.oracletutoring.ca\">Oracle Tutoring by Jack and Diane,<\/a> Cambpell River, BC.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The tutor shows an example of a projection from one dimension into another. Let&#8217;s imagine we have the vector v 3 2 0 which we want to project in the direction of w 3 -1 0 The projection matrix, Pw, &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"more-link\" href=\"https:\/\/www.oracletutoring.ca\/blog\/linear-algebra-projecting-a-vector-along-another-one-matrix-method\/\"> <span class=\"screen-reader-text\">Linear algebra:  projecting a vector along another one, matrix method.<\/span> Read More &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[302,3,2004],"tags":[2023,2024],"class_list":["post-18706","post","type-post","status-publish","format-standard","hentry","category-linear-algebra","category-math","category-matrix-algebra","tag-projection","tag-projection-matrix"],"_links":{"self":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/18706","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/comments?post=18706"}],"version-history":[{"count":14,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/18706\/revisions"}],"predecessor-version":[{"id":18720,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/18706\/revisions\/18720"}],"wp:attachment":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/media?parent=18706"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/categories?post=18706"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/tags?post=18706"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}