Math 375 - 10/18 - Shawn Zhong

$Theorem • V,W: vector spaces • x_1,…,x_n: basis for V • For any w_1,…,w_n∈W • There is a unique linear map T:V→W • s.t. {█(T(x_1 )=w_1@⋮@T(x_n )=w_n )┤ • v∈W⇒∃c_1,…,c_n∈R • s.t. v=c_1 x_1+…+c_n x_n • T(v)=c_1 w_1+…+c_n w_n • Linear map can be determined only by operations on basis Question 1 • Requirement ○ T: R2→R3 ○ dim⁡(range(T))=1 • Example ○ T(x,y)=(x,0,0) ○ T(x,y)=(0,y,y) ○ T(x,y)=(0,x+3y,−2x−6y) Question 2 • Requirement ○ T: R2→R2 ○ S: R2→R2 ○ ST=−TS • Example ○ T(x,y)=(−y,x) ○ S(x,y)=(−x,y) Question 3 • Requirement ○ T: R3→R3 ○ T^2≠0 ○ T^3≠0 • Example ○ T(x,y,z)=(0,x,y) Question 4 • Requirement ○ T: R2→R2 ○ T maps the unit square to the parallgram below ○ T(0,0)=(0,0) ○ T(1,0)=(2,0) ○ T(0,1)=(1,1) ○ T(1,1)=(3,0) • Example ○ T(x,y)=(2x+y,y) ○ T(x,y)=(2y+x,x) Question 5 • Requirement ○ T: R3→R3 ○ T(x,0,0)=(2x,0,0) ○ T^3 (0,a,b)=(0,a,b) • Example ○ T(x,y,z)=(2x,y,z) Question 6 • Requirement ○ T: R2→R2 ○ T(1,0)=(1,0) ○ {(x,y),T(x,y)} is independent whenever y≠0 • Example ○ T(x,y)=(x+y,y) Question 7 • Requirement ○ T: R2→R3 ○ T is injective ○ dim⁡(range(T))=1 • Example ○ Impossible " t • --- .... ice (1.1) T (1.1) (3.1) I _ _ _ _ _ " - _ ~) > > (2.0)’ class=’alignnone size-full wp-image-2407 ‘ src=’https://shawnzhong.com/wp-content/uploads/2017/11/img_59fd389386cb7.png’ /></p> </div> <div class="entry-footer"> </div> </div> </article> <div class="post-attributes">  <div class="related-posts"> <h3>Related Posts</h3> <div class="multi-carousel onetone-related-posts owl-carousel owl-theme"> <div class="owl-item"> <div class="post-grid-box"> <div class="img-box figcaption-middle text-center from-left fade-in"> <a href="http://3.145.115.83/2017/10/26/math-375/"> <img src="http://3.145.115.83/wp-content/uploads/2017/10/calculus-applied_378x225.jpg" class="feature-img"/> <div class="img-overlay"> <div class="img-overlay-container"> <div class="img-overlay-content"> <i class="fa fa-link"></i> </div> </div> </div> </a> </div> <div class="img-caption"> <a href="http://3.145.115.83/2017/10/26/math-375/"><h4>Math 375</h4></a> <ul class="entry-meta"> <li class="entry-date"><i class="fa fa-calendar"></i>October 26, 2017</li> <li class="entry-author"><i class="fa fa-user"></i>Shawn</li> </ul> </div> </div> </div> </div> </div>   <div class="comments-area text-left">  <div id="respond" class="respondbg"> <div id="respond" class="comment-respond"> <h3 id="reply-title" class="comment-reply-title">Leave a Reply <small><a rel="nofollow" id="cancel-comment-reply-link" href="/2017/10/27/1018/#respond" style="display:none;">Cancel reply</a></small></h3><form action="http://3.145.115.83/wp-comments-post.php" method="post" id="commentform" class="comment-form" novalidate><p class="comment-notes">Your email address will not be published.</p><div class="clear"></div><p class="form-allowed-tags"></p> <section class="comment-form-comment form-group"><div id="comment-textarea"><textarea id="comment" name="comment" placeholder="Message" cols="45" rows="8" class="textarea-comment form-control" aria-required="true"></textarea></div></section><div class="row"><section class="comment-form-author form-group col-md-4"><input id="author" class="input-name form-control" name="author" placeholder="Name" type="text" value="" size="30" aria-required=$

Shawn Zhong

Shawn Zhong

Math 375 – 10/18

Math 375 – 10/18

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