{"id":3544,"date":"2005-09-28T00:17:00","date_gmt":"2005-09-27T15:17:00","guid":{"rendered":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/?p=3544"},"modified":"2021-08-12T11:56:33","modified_gmt":"2021-08-12T02:56:33","slug":"la2k5fallpractice0920","status":"publish","type":"post","link":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/2005\/09\/28\/la2k5fallpractice0920\/","title":{"rendered":"LA2k5FallPractice0920"},"content":{"rendered":"

=<\/code> 9\/20 \uc5f0\uc2b5 \ub0b4\uc6a9 =<\/code> ”’\uace0\uccd0\ub193\uc558\uc5b4\uc694. – \uae40\uc601\uc6b1”’ <\/p>\n

\\[ 16. ~\\text{ Suppose that u,v, and w are vectors such that} =2, = -3, = 5,|u| = 1, \\] \\[ |v|=2, |w|=7 \\] ~$$ ~<\/code>\\text{ Evaluate the given expression. }~~ \\text{(a)} ~< u+v,v+w> ~~~<\/code> \\text{(b)}~~ < 2v-w ,3u+2w> $$ <\/p>\n

Ans of (a) : \\[ = + + + = 2 + 5 +4 -3 = 8 . \\] <\/p>\n

Ans of (b) : \\[ = 6 +4 -3 -2 = 12 -12 -15-98 = -113 . \\] <\/p>\n

\\[ 21. ~\\text{ Show that the following identity holds for vectors in any inner product space.} \\] <\/p>\n

\\[ = {\\frac14} |u+v|^2 – \\frac14 |u-v|^2 \\] <\/p>\n

Ans : \\[ Since~~ |u+v|^2 = = + + + \\] \\[ Since~~ |u-v|^2 = = – – + \\] \\[ Thus~~~ |u+v|^2 – |u-v|^2 = 4 . \\] <\/p>\n

\\[ 24. ~\\text{ Prove : If } \\text{is the Euclidean inner product on}~ R^n ,\\text{ and if}~ A~ \\text{is an} \\] n\u00d7n \\[ \\text{matrix , then} \\] \\[ = \\] <\/p>\n

Ans : \\[ \\text{Since} = (Av)^T u = v^T A^T u = v^T (A^T u) = . \\] <\/p>\n

\\[ 27. ~\\text{ Use the inner pruduct to compute} \\] \\[ <\/p>\n

\\text{for the vectors p=p(x) and q=q(x) in} P_3 \\] <\/p>\n

\\[ ~~~~~~~~~~~~~~~<\/p>\n

= \\int_{-1}^1~~ p(x)q(x)~ dx \\] <\/p>\n

Ans : Check first this is inner product. <\/p>\n

\u2460 \\[ <\/p>\n

= \\int_{-1}^1~~ p(x)q(x)~ dx ~~~~~~ = \\int_{-1}^1~~q(x)p(x) ~ dx \\] <\/p>\n

\u2461 \\[ <\/p>\n

= \\int_{-1}^1~~ p(x)[q(x)+r(x)]~ dx = \\int_{-1}^1~~p(x)q(x)~dx ~~+\\int_{-1}^1~~p(x)r(x)~dx ~~then ~ <\/p>\n

= <\/p>\n

+ <\/p>\n

~ \\] <\/p>\n

\u2462 \\[ = \\int_{-1}^1~~ [kp(x)]q(x)~ dx ~~=k\\int_{-1}^1~~ p(x)q(x)~ dx ~~\\] <\/p>\n

\u2463 \\[ <\/p>\n

= \\int_{-1}^1~~ p(x)^2~ dx~ \\geq 0 ~~ \\text{Since} ~~ p(x)^2 \\geq 0 ~~~ \\text{and}~~ <\/p>\n

=0 ~~\\text{iff}~~ p(x)=0 \\] <\/p>\n

And we sloved the problem ”’31”’. <\/p>\n

\\[ 19. \\text{ Let V be an inner product space. Show that if}~ u~\\text{ and}~ v \\text{ are orthogonal unit vectors in}~ V, \\text{ then}~ |u-v| = \\] <\/p>\n

Ans : \\[ \\text{ Since }~ u \\text{ and}~ v \\text{ are orthogonal, then} \\] \\[ = o ~\\text{and Since}~ u \\text{ and}~ v \\text{ are a unit then}~ |u|=|v|=1 \\] \\[ \\text{Thus}~ |u-v|^2 = = – 2 + = 1+1 =2 . \\] <\/p>\n

\\[ 24. \\text{ Prove the following generalization of Theorem 6.2.4. If }~v_1, v_2 , \\dots , v_r \\text{ are pairwise orthogonal vectors in an inner product space in V,} \\] \\[ \\text{ then} ~ |v_1 + v_2 + \\dots + v_r |^2 = |v_1|^2 + \\dots +|v_r|^2 \\] <\/p>\n

Ans : \\[ \\text{ Since}~ v_i’s \\text{ are pairwise orthogonal, then}~ =0 \\text{ if } i\\neq j . \\text{thus proved.} \\] <\/p>\n","protected":false},"excerpt":{"rendered":"

= 9\/20 \uc5f0\uc2b5 \ub0b4\uc6a9 = ”’\uace0\uccd0\ub193\uc558\uc5b4\uc694. – \uae40\uc601\uc6b1”’ \\[ 16. ~\\text{ Suppose that u,v, and w are vectors such that} =2, = -3, = 5,|u| = 1, \\] \\[ |v|=2, |w|=7 \\] ~$$ ~\\text{ Evaluate the given expression. }~~ \\text{(a)} ~< u+v,v+w> ~~~ \\text{(b)}~~ < 2v-w ,3u+2w> $$ Ans of (a) : \\[ = + … \ub354 \uc77d\uae30<\/a><\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ngg_post_thumbnail":0,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-3544","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"distributor_meta":false,"distributor_terms":false,"distributor_media":false,"distributor_original_site_name":"\uae40\uc601\uc6b1","distributor_original_site_url":"https:\/\/mathematicians.korea.ac.kr\/ywkim","push-errors":false,"_links":{"self":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts\/3544","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/comments?post=3544"}],"version-history":[{"count":1,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts\/3544\/revisions"}],"predecessor-version":[{"id":3545,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts\/3544\/revisions\/3545"}],"wp:attachment":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/media?parent=3544"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/categories?post=3544"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/tags?post=3544"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}