[wiki:\uc120\ud615\ub300\uc218\uc219\uc81c: \uc704\ub85c] <\/p>\n
B. Duality, dual basis. <\/p>\n
%\\resume{enumerate} <\/p>\n
\\begin{enumerate}%\\setcounter{enumi}{7}
\n\\item
\nSuppose that for each $x$ in $\\mathcal{P}$
\nthe function $y$ is defined by
\n \\begin{enumerate}
\n \\item
\n $y(x)=\\int_{-1}^2 x(t)\\,dt$
\n \\item
\n $y(x)=\\int_0^2 (x(t))^2\\,dt$
\n \\item
\n $y(x)=\\int_0^1 t^2x(t)\\,dt$
\n \\item
\n $y(x)=\\int_0^1 x(t^2)\\,dt$
\n \\item
\n $y(x)=\\dfrac{dx}{dt}$
\n \\item
\n $y(x)=\\dfrac{d^2x}{dt^2}\\bigg|_{t=1}$
\n \\end{enumerate}<\/p>\n
In which of these cases is \\(y\\) a linear function? <\/p>\n
\\item If \\(y\\) is a non-zero linear function on a vector space \\(V\\), and if \\(\\alpha\\) is an arbitrary scalar, does there necessarily exist a vector \\(x\\) in \\(V\\) such that \\(y(x)=\\alpha\\)? <\/p>\n
\\item Prove that if \\(y\\) and \\(z\\) are linear functions (on the same vector space) such that \\(y(x)=0\\) whenever \\(z(x)=0\\), then there exists a scalar \\(\\alpha\\) such that \\(y=\\alpha z\\). (Hint: if \\(z(x_0)\\neq0\\), write \\(\\alpha=y(x_0)\/z(x_0)\\).) <\/p>\n
\\item Suppose that \\(m<n\\) and that \\(y_1, \\dots, y_m\\) are linear functionals on an $n$-dimensional vector space \\(V\\). Under what conditions on the scalars \\(\\alpha_1, \\dots, \\alpha_m\\) is it true that there exists a vector \\(x\\) in \\(V\\) such that \\(y_j(x)=\\alpha_j\\) for \\(j=1,\\dots,m\\)? <\/p>\n
What does this result say about the solutions of linear equations? <\/p>\n
\\end{enumerate} <\/p>\n<\/div>\n<\/div>\n
[wiki:\uc120\ud615\ub300\uc218\uc219\uc81c: \uc704\ub85c] <\/p>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"
[wiki:\uc120\ud615\ub300\uc218\uc219\uc81c: \uc704\ub85c] \ub450\ubc88\uc9f8 \uc219\uc81c\uc785\ub2c8\ub2e4. B. Duality, dual basis. %\\resume{enumerate} \\begin{enumerate}%\\setcounter{enumi}{7} \\item Suppose that for each $x$ in $\\mathcal{P}$ the function $y$ is defined by \\begin{enumerate} \\item $y(x)=\\int_{-1}^2 x(t)\\,dt$ \\item $y(x)=\\int_0^2 (x(t))^2\\,dt$ \\item $y(x)=\\int_0^1 t^2x(t)\\,dt$ \\item $y(x)=\\int_0^1 x(t^2)\\,dt$ \\item $y(x)=\\dfrac{dx}{dt}$ \\item $y(x)=\\dfrac{d^2x}{dt^2}\\bigg|_{t=1}$ \\end{enumerate} In which of these cases is \\(y\\) a linear function? \\item If \\(y\\) … \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-3844","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\/3844","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=3844"}],"version-history":[{"count":1,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts\/3844\/revisions"}],"predecessor-version":[{"id":3845,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts\/3844\/revisions\/3845"}],"wp:attachment":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/media?parent=3844"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/categories?post=3844"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/tags?post=3844"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}