D. Determinants, Jordan form, Euclidean structures. <\/p>\n
\\begin{enumerate}
\n\\item
\nIf $A$ and $B$ are linear transformations such that
\n$AB=0$, $A\\neq0$, $B\\neq0$, then
\n$\\det A=\\det B=0$.<\/p>\n
\\item
\nProve the following and see if the converse is true.
\n \\begin{enumerate}
\n \\item
\n If $A$ and $B$ are similar, then $\\det A=\\det B$.
\n \\item
\n If $A$ and $B$ are similar, then $A$ and $B$ have the
\n same characteristic polynomial.
\n \\item
\n If $A$ and $B$ have the same characteristic polynomial,
\n then $\\det A=\\det B$.
\n \\end{enumerate}<\/p>\n
\\item On \\(\\mathcal{P}_n\\), the space of polynomials \\(p(t)\\) of degree \\(\\leq n\\), consider the differential operator \\(D=\\partial\/\\partial t\\). Find the Jordan form of \\(D\\). Find the minimal polynomial of \\(D\\). (Hint: Use the standard basis \\(1,t,t^2,\\dots,t^n\\) to compute the matrix representation for \\(D^k\\).) <\/p>\n
\\item Read the theorem 12, p.{} 57 of our textbook.(It is not necessary to read the proof.) Then show that the following matrices are similar. \\[ <\/p>\n
\\begin{pmatrix}
\n0&1&\\alpha \\\\ 0&0&1 \\\\ 0&0&0
\n\\end{pmatrix}<\/p>\n
\\quad <\/p>\n
\\begin{pmatrix}
\n0&1&0 \\\\ 0&0&1 \\\\ 0&0&0
\n\\end{pmatrix}<\/p>\n
\\] <\/p>\n
\\item For \\(x=(x_1,x_2)\\in \\mathbb{R}^2\\) define \\[ \\|x\\|:=\\max \\{ |x_1|, |x_2| \\}. \\] Check that this function satisfy triangle inequality. And prove that there is no innerproduct \\((~,~)\\) in \\(\\mathbb{R}^2\\) so that \\(\\|x\\|^2=(x,x)\\). <\/p>\n
\\item Let \\(A\\) be a self-adjoint linear transformation. Show that the following formula holds: \\[ (Ax,y)+(Ay,x) = (A(x+y),(x+y)) – (Ax,x) – (Ay,y). \\] Use this formula to show that, if \\((Ax,x)=0\\) for all \\(x\\) then \\(A=0\\). <\/p>\n
\\end{enumerate} <\/p>\n","protected":false},"excerpt":{"rendered":"
D. Determinants, Jordan form, Euclidean structures. \\begin{enumerate} \\item If $A$ and $B$ are linear transformations such that $AB=0$, $A\\neq0$, $B\\neq0$, then $\\det A=\\det B=0$. \\item Prove the following and see if the converse is true. \\begin{enumerate} \\item If $A$ and $B$ are similar, then $\\det A=\\det B$. \\item If $A$ and $B$ are similar, then … \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-3850","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\/3850","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=3850"}],"version-history":[{"count":1,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts\/3850\/revisions"}],"predecessor-version":[{"id":3851,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts\/3850\/revisions\/3851"}],"wp:attachment":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/media?parent=3850"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/categories?post=3850"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/tags?post=3850"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}