{"id":178,"date":"2020-07-23T08:50:23","date_gmt":"2020-07-23T08:50:23","guid":{"rendered":"http:\/\/www.risanp.com\/notes\/?p=178"},"modified":"2020-07-23T08:55:51","modified_gmt":"2020-07-23T08:55:51","slug":"axlers-linear-algebra-done-right-exercises-1c-24","status":"publish","type":"post","link":"https:\/\/www.risanp.com\/notes\/?p=178","title":{"rendered":"Axler’s Linear Algebra Done Right, Exercise 1C-24"},"content":{"rendered":"\n

<\/p>\n\n\n\n

Problem<\/strong>
A function \"f:\mathbb{R} \rightarrow \mathbb{R}\" is called even if

  <\/span>   <\/span>\"\[ f(-x) = f(x)\]\"<\/p>
for all \"x \in \mathbb{R}\". A function \"f:\mathbb{R} \rightarrow \mathbb{R}\" is called odd if

  <\/span>   <\/span>\"\[ f(-x) = -f(x)\]\"<\/p>
for all \"x \in \mathbb{R}\". Let \"U_e\" denote the set of real-valued even functions on \"\mathbb{R}\" and let \"U_o\" denote the set of real-valued odd functions on \"\mathbb{R}\". Show that:

  <\/span>   <\/span>\"\[ \mathbb{R}^\mathbb{R}=U_e \oplus U_o\]\"<\/p>

Proof.<\/strong><\/em> To show that \"U_e + U_o\" is a direct sum, it is sufficient to prove that \"U_e \cap U_o = {0}\". Let \"f \in U_e \cap U_o\". Hence, \"f(x) = f(-x) = -f(-x)\" for all \"x \in R\", which only happens for \"f(x) = 0\". Hence, \"U_e + U_o\" is a direct sum.

Let \"g \in \mathbb{R}^\mathbb{R}\". The equation \"\mathbb{R}^\mathbb{R}=U_e \oplus U_o\" means that \"g\" can be written as the sum of an odd function \"o\" and an even function \"e\". Or in other words, for all \"x \in \mathbb{R}\":
<\/a>

(1) <\/span>   <\/span>\"\begin{equation*} g(x) = o(x) + e(x)\end{equation*}\"<\/p><\/p>\n\n\n\n

and also:
<\/a>

(2) <\/span>   <\/span>\"\begin{equation*} g(-x) = o(-x) + e(-x)\end{equation*}\"<\/p>
By the definition of the odd and even functions, (2<\/a>) can be written as:
<\/a>

(3) <\/span>   <\/span>\"\begin{equation*} g(-x) = -o(x) + e(x)\end{equation*}\"<\/p>
The system of linear equations (1<\/a>) and (3<\/a>) has a unique solution \"o(x) = (g(x) - g(-x)) / 2\" and \"e(x) = (g(x) + g(-x)) / 2\". Hence, \"g\" can always be uniquely written as the sum of an odd function and an even function. \"\qed\"<\/p>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

ProblemA function is called even if     for all . A function is called odd if     for all . Let denote the set of real-valued even functions on and let denote the set of real-valued odd functions … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[10,9,6],"tags":[],"class_list":["post-178","post","type-post","status-publish","format-standard","hentry","category-axlers-linear-algebra-done-right","category-linear-algebra","category-mathematics"],"_links":{"self":[{"href":"https:\/\/www.risanp.com\/notes\/index.php?rest_route=\/wp\/v2\/posts\/178"}],"collection":[{"href":"https:\/\/www.risanp.com\/notes\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.risanp.com\/notes\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.risanp.com\/notes\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.risanp.com\/notes\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=178"}],"version-history":[{"count":10,"href":"https:\/\/www.risanp.com\/notes\/index.php?rest_route=\/wp\/v2\/posts\/178\/revisions"}],"predecessor-version":[{"id":190,"href":"https:\/\/www.risanp.com\/notes\/index.php?rest_route=\/wp\/v2\/posts\/178\/revisions\/190"}],"wp:attachment":[{"href":"https:\/\/www.risanp.com\/notes\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=178"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.risanp.com\/notes\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=178"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.risanp.com\/notes\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=178"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}