{"id":248163,"date":"2025-07-08T08:00:23","date_gmt":"2025-07-08T08:00:23","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=248163"},"modified":"2025-07-08T08:00:26","modified_gmt":"2025-07-08T08:00:26","slug":"perform-the-gram-schmidt-process-on-the-following-sequence-of-vectors","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/08\/perform-the-gram-schmidt-process-on-the-following-sequence-of-vectors\/","title":{"rendered":"Perform the Gram-Schmidt process on the following sequence of vectors"},"content":{"rendered":"\n
Perform the Gram-Schmidt process on the following sequence of vectors<\/pre>\n\n\n\n
\"\"<\/figure>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n
Correct Answer<\/strong><\/p>\n\n\n\n
The orthogonal set of vectors obtained from the Gram-Schmidt process is:<\/p>\n\n\n\n
{ [-6, -3, -6], [1, 2, -2], [-2, 2, 1] }<\/p>\n\n\n\n
This corresponds to filling the boxes as:
First vector:
-6
-3
-6<\/p>\n\n\n\n
Second vector:
1
2
-2<\/p>\n\n\n\n
Third vector:
-2
2
1<\/p>\n\n\n\n
\n\n\n\n
Explanation<\/strong><\/p>\n\n\n\n
The Gram-Schmidt process is a standard method in linear algebra used to convert a set of linearly independent vectors into an orthogonal set that spans the same subspace. An orthogonal set is one where every pair of vectors is perpendicular, meaning their dot product is zero. This process is essential because working with orthogonal vectors simplifies many calculations. The method works sequentially, taking each vector from the original set and removing its components that are parallel to the previously generated orthogonal vectors.<\/p>\n\n\n\n
Let the original vectors be v\u2081 = [-6, -3, -6], v\u2082 = [-3, 0, -6], and v\u2083 = [-2, -1, 7]. We will find the new orthogonal vectors u\u2081, u\u2082, and u\u2083.<\/p>\n\n\n\n
    \n
  1. First Vector:<\/strong>\u00a0The procedure begins by setting the first vector of our new orthogonal set, u\u2081, equal to the first vector of the original set, v\u2081.
    u\u2081 = v\u2081 = [-6, -3, -6].<\/li>\n\n\n\n
  2. Second Vector:<\/strong>\u00a0To find the second orthogonal vector, u\u2082, we take the second original vector, v\u2082, and subtract its projection onto u\u2081. The formula is u\u2082 = v\u2082 – [(v\u2082 \u22c5 u\u2081) \/ (u\u2081 \u22c5 u\u2081)] * u\u2081.
    First, we compute the necessary dot products:
    v\u2082 \u22c5 u\u2081 = (-3)(-6) + (0)(-3) + (-6)(-6) = 18 + 0 + 36 = 54.
    u\u2081 \u22c5 u\u2081 = (-6)\u00b2 + (-3)\u00b2 + (-6)\u00b2 = 36 + 9 + 36 = 81.
    The projection coefficient is 54\/81, which simplifies to 2\/3.
    So, u\u2082 = [-3, 0, -6] – (2\/3) * [-6, -3, -6] = [1, 2, -2].<\/li>\n\n\n\n
  3. Third Vector:<\/strong>\u00a0For the third orthogonal vector, u\u2083, we take the third original vector, v\u2083, and subtract its projections onto both u\u2081 and u\u2082. The formula is u\u2083 = v\u2083 – [(v\u2083 \u22c5 u\u2081) \/ (u\u2081 \u22c5 u\u2081)] * u\u2081 – [(v\u2083 \u22c5 u\u2082) \/ (u\u2082 \u22c5 u\u2082)] * u\u2082.
    We calculate the new dot products:
    v\u2083 \u22c5 u\u2081 = (-2)(-6) + (-1)(-3) + (7)(-6) = -27.
    v\u2083 \u22c5 u\u2082 = (-2)(1) + (-1)(2) + (7)(-2) = -18.
    u\u2082 \u22c5 u\u2082 = 1\u00b2 + 2\u00b2 + (-2)\u00b2 = 9.
    The projection coefficients are -27\/81 = -1\/3 and -18\/9 = -2.
    The calculation is u\u2083 = [-2, -1, 7] – (-1\/3) * [-6, -3, -6] – (-2) * [1, 2, -2], which simplifies to u\u2083 = [-2, 2, 1].<\/li>\n<\/ol>\n\n\n\n
    The final set of orthogonal vectors is {[-6, -3, -6], [1, 2, -2], [-2, 2, 1]}.<\/p>\n\n\n\n
    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"
    Perform the Gram-Schmidt process on the following sequence of vectors The Correct Answer and Explanation is: Correct Answer The orthogonal set of vectors obtained from the Gram-Schmidt process is: { [-6, -3, -6], [1, 2, -2], [-2, 2, 1] } This corresponds to filling the boxes as:First vector:-6-3-6 Second vector:12-2 Third vector:-221 Explanation The Gram-Schmidt […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-248163","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/248163","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=248163"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/248163\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=248163"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=248163"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=248163"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}