\n

\n Let A:=[1 1 1,1 -2 0, 0 3 1] and y0 =[1 5 0]. We seek to find the column vector x0 \u2208R3 of smallest\n <\/p>\n

\n 101 0\n <\/p>\n

\n norm such that Ax0 is of minimal distance from y0 , where the norm on R3 is the usual Euclidean norm coming from the dot product.\n <\/p>\n

\n To keep the ideas straight, you should keep in mind the facts that as a linear function R3 \u2192 R3 , the kernel of the linear function L(x) := Ax is the null space Nul(A) and the image of L is the column space Col(A) of A.\n <\/p>\n

\n (a) Use row and column operations to put A in standard form, and hence show that the nullity of A is 1.\n <\/p>\n

\n (b) We need to find the orthogonal projection of y0 onto the column space of A, but the latter space has dimension 2 so it is easier to find the orthogonal projection of y0 onto (Nul(At)) . Find a vector spanning Nul(At), and use this to find the orthogonal projection y1 of y0 on Col(A).\n <\/p>\n

\n (c) Find x1 \u2208 R3 satisfying Ax2 = y.\n <\/p>\n

\n (d) Find a vector spanning Nul(A), and by calculating the orthogonal projection of x1 onto Nul(A) , find the vector x0 \u2208 R3 of minimal norm for which Ax0 is of minimal possible distance from y0.\n <\/p>\n

1 1 0 11 \u20142 3 and yo = 5 . We seek to \ufb01nd the column vector x0 E R3 of smallest1 0 1 0 norm such that Axe is of minimal distance from yo , Where the norm on R3 is the usual Euclidean Let A :2 norm coming from the dot product. To keep the ideas straight, you should keep in mind the factsthat as a linear function R3 \u2014&gt; R3 , the kernel of the linear function L(x) :2 Ax is the null spaceN ul(A) and the image of L is the column space Col (A) of A.\n<\/p><\/div>\n","protected":false},"excerpt":{"rendered":"

Let A:=[1 1 1,1 -2 0, 0 3 1] and y0 =[1 5 0]. We seek to find the column vector x0 \u2208R3 of smallest 101 0 norm such that Ax0 is of minimal distance from y0 , where the norm on R3 is the usual Euclidean norm coming from the dot product. To keep […]<\/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\/228505"}],"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=228505"}],"version-history":[{"count":0,"href":"https:\/\/qualityassignments.net\/wp-json\/wp\/v2\/posts\/228505\/revisions"}],"wp:attachment":[{"href":"https:\/\/qualityassignments.net\/wp-json\/wp\/v2\/media?parent=228505"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/qualityassignments.net\/wp-json\/wp\/v2\/categories?post=228505"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/qualityassignments.net\/wp-json\/wp\/v2\/tags?post=228505"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}