[wiki:\uc120\ud615\ub300\uc218\uc219\uc81c: \uc704\ub85c] <\/p>\n
A. Vector space, subspace, linear dependence, basis, dimension, direct sum, quotient space. <\/p>\n
\\begin{enumerate}
\n\\item
\nLet $\\mathbb{Q}(\\sqrt{2})$ be the set of all real numbers
\nof the form $\\alpha+\\beta\\sqrt{2}$, where
\n$\\alpha$ and $\\beta$ are rational.
\n \\begin{enumerate}
\n \\item
\n Is $\\mathbb{Q}(\\sqrt{2})$ a field?
\n \\item
\n What if $\\alpha$ and $\\beta$ are required to be integers?
\n \\end{enumerate}<\/p>\n
\\item Review the problems from Spring semester regarding the examples of a set with operations \\(+\\) and \\(\\cdot\\) which form vector spaces and which do not. <\/p>\n
\\item Prove that if \\(\\mathbb{R}\\) is considered as a rational vector space, i.e., a vector space with the rational number field \\(\\mathbb{Q}\\) as the scalar field, then a necessary and sufficient condition that the vectors \\(1\\) and \\(\\xi\\) in \\(\\mathbb{R}\\) be linearly independent is that the real number \\(\\xi\\) be irrational. <\/p>\n
\\item Is it true that if \\(x,y,z\\) are linearly independent vectors, then so also are \\(x+y\\), \\(y+z\\) and \\(z+x\\)? <\/p>\n
\\item <\/p>\n
\\begin{enumerate}
\n\\item
\nProve or disprove the followings(PODF):
\nThe vectors $(\\xi_1,\\xi_2)$ and $(\\eta_1,\\eta_2)$
\nin $\\mathbb{C}^2$ are linearly dependent if and only if
\n$\\xi_1\\eta_2=\\xi_2\\eta_1$.<\/p>\n
\\item
\nFind a similar necessary and sufficient condition for the
\nlinear dependence of two vectors in $\\mathbb{C}^3$.<\/p>\n
\\item
\nIs there a set of three linearly independent vectors in
\n$\\mathbb{C}^2$?
\n\\end{enumerate}<\/p>\n
\\item Is the set \\(\\mathbb{R}\\) of real numbers a finite-dimensional vector space over the field \\(\\mathbb{Q}\\) of rational numbers? (Knowing somthing about cardinal numbers will help.) <\/p>\n
\\item If \\(M\\) and \\(N\\) are subspaces of a vector space \\(V\\), and if every vector in \\(V\\) belongs either to \\(M\\) or \\(N\\) (or both), then either \\(M=V\\) or \\(N=V\\) (or both). <\/p>\n
\\item Consider the quotient spaces obtained by reducing the space \\(\\mathcal{P}\\) of polynomials modulo various subspaces. If \\(M=\\mathcal{P}_n\\), is \\(\\mathcal{P}\/M\\) finite-dimensional? What if \\(M\\) is the subspace consisting of all even polynomials? What if \\(M\\) is the subspace consistiong of all polynomials divisible by \\(x_n\\) (where \\(x_n(t)=t^n\\))? <\/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] \uccab\ubc88\uc9f8 \uc219\uc81c\uc785\ub2c8\ub2e4. A. Vector space, subspace, linear dependence, basis, dimension, direct sum, quotient space. \\begin{enumerate} \\item Let $\\mathbb{Q}(\\sqrt{2})$ be the set of all real numbers of the form $\\alpha+\\beta\\sqrt{2}$, where $\\alpha$ and $\\beta$ are rational. \\begin{enumerate} \\item Is $\\mathbb{Q}(\\sqrt{2})$ a field? \\item What if $\\alpha$ and $\\beta$ are required to be integers? \\end{enumerate} … \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-3842","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\/3842","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=3842"}],"version-history":[{"count":1,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts\/3842\/revisions"}],"predecessor-version":[{"id":3843,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts\/3842\/revisions\/3843"}],"wp:attachment":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/media?parent=3842"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/categories?post=3842"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/tags?post=3842"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}