[wiki:\uc120\ud615\ub300\uc218\uc219\uc81c: \uc704\ub85c] <\/p>\n
C. Linear Transformations and Their Matrices <\/p>\n
\\begin{enumerate}
\n\\item
\nIf $A$ and $B$ are linear transformations (on the same vector space),
\nthen a necessary and sufficient condition that both $A$ and $B$ be invertible is that
\nboth $AB$ and $BA$ be invertible.
\n\\item
\nIf $A$ and $B$ are linear transformations on a finite-dimensional vector space,
\nand if $AB=I$, then both $A$ and $B$ are invertible.
\n\\item
\nIf $A$ and $B$ are linear transformations (on the same vector space) and if
\n$AB=I$, then $A$ is called a \\textem{left inverse} of $B$ and
\n$B$ is called a \\textem{right inverse} of $A$.
\nProve that if $A$ has exactly one right inverse, say $B$, then $A$ is invertible.
\n(Hint: consider $BA+B-I$.)
\n\\item
\nProve that if $e_1$, $e_2$, and $e_3$ are the complex matrices
\n$$
\n\\begin{pmatrix} 0 & 1 \\\\ -1 & 0 \\end{pmatrix},
\n\\quad
\n\\begin{pmatrix} 0 & i \\\\ i & 0 \\end{pmatrix},
\n\\quad
\n\\begin{pmatrix} i & 0 \\\\ 0 & -i \\end{pmatrix}
\n$$
\nrespectively (where $i=\\sqrt{-1}$), then
\n$e_1^2=e_2^2=e_3^2=-I$,
\n$e_1e_2=-e_2e_1=e_3$,
\n$e_2e_3=-e_3e_2=e_1$,
\nand
\n$e_3e_1=-e_1e_3=e_2$.
\n\\item
\n{\uad50\uacfc\uc11c 19\ucabd Exercise 3}.
\n\\item
\nLet $\\mathcal{L}$ be the space of linear maps from $U^n$ to $V^m$.
\nAnd let $\\mathcal{M}$ be the space of $m\\times n$ matrices.
\nChoose and fix a basis for $U$ and a basis for $V$.
\nWith respect to these bases, to each linear map $T\\in \\mathcal{L}$
\ncan we associate an $m\\times n$ matrix $A_T$ (as was explained in the spring semester).
\nLet us denote this association by $\\Phi$.
\nShow that this map $\\Phi$ is a vector space isomorphism which also satisfies
\nthe following identity.
\n$$
\n\\Phi(S\\circ T) = \\Phi(S)\\Phi(T)\\quad\\text{or}\\quad
\nA_{S\\circ T}=A_S A_T
\n$$
\nand
\n$$
\n\\Phi(T^{-1}) = \\Phi(T)^{-1}\\quad \\text{or} \\quad
\nA_{T^{-1}} = A_T^{-1}
\n$$<\/p>\n
\\end{enumerate}<\/p>\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] C. Linear Transformations and Their Matrices \\begin{enumerate} \\item If $A$ and $B$ are linear transformations (on the same vector space), then a necessary and sufficient condition that both $A$ and $B$ be invertible is that both $AB$ and $BA$ be invertible. \\item If $A$ and $B$ are linear transformations on a finite-dimensional vector … \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-3846","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\/3846","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=3846"}],"version-history":[{"count":1,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts\/3846\/revisions"}],"predecessor-version":[{"id":3847,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts\/3846\/revisions\/3847"}],"wp:attachment":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/media?parent=3846"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/categories?post=3846"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/tags?post=3846"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}