Worksheet 13 (March 3) DIS 119/120 GSI Xiaohan Yan 1 Review DEFINITIONS • coordinate mapping; • matrix of linear transformation relative to bases on domain and codomain; METHODS AND IDEAS Theorem 1. Any two vector spaces of the same dimension are isomorphic, since any vector space of dimension n is isomorphic to R n under the coordinate mapping under a basis. Note that vector spaces of di↵erent dimensions can never be isomorphic. This is because isomorphism preserves linear independence and spanning property, and thus always sends a basis to a basis. But bases of vector spaces of di↵erent dimensions contain di↵erent number of vectors. 2 Problems Example 1. True or false. In the last three statements, S denotes the vector space of all smooth (infinitely di↵erentiable) functions over r0, 1s. In other words, you do not need to worry about di↵erentiability of elements of S ( ) There exists a basis B of P 2 such that 1 ` x has coordinate p1, 1, 1q T while x ` x 2 has coordinates p´3, ´3, ´3q. ( ) Let V be a 5-dimensional vector space and v 1 , v 2 be two linearly inde- pendent vectors in V , then there exists three other vectors u 1 , u 2 , u 3 in V such that tv 1 , v 2 , u 1 , u 2 , u 3 u is a basis of V . ( ) Let W be a 2-dimensional vector space and v 1 , v 2 , v 3 , v 4 , v 5 be five vectors in W , then we can take two of these vectors to form a basis of W . 1 Recall T.IR 11pm basis B of V bi ibis Thi't _A P V Bpi isomorphism A Tei Tien's c bit tabi al e r ai Kil T vw B Sbi a III T V W T BE e ki aiy BR a ke llwtc p.TT e lv7B1Rn slRmlv7BgnenToymatixlTiiYe BITIC bide i Fib'Re E vibituabi't v In Min Thi's Tiv.io i rnb i uTibTii vnTlbnT ITNYK r.fi Dc i tvnTTlbi'Do Lus Hibiya Embiid Rns B T p B 1123 iso T r FIB F T I memorize the statement i i l this statement uouidbetrueifspar.su ivIg W3 Tiu7tTiwy F any f d vector Idea wecanjustcheckthisfoyps.dspauisiso.to 113in Ingen eral the Euclidean 354in Eep We can alwayscomplete T two t.I.co umnvectorsotthesamed.m.asetofL 2 vectorsinV xp column vectors spanks 34 1 2 0 40 10 to a basis u 7.5 pivot columns impossible we can always remove redundancy we can determine pivot columns by tow reductions from aset of spanningvectossofu vi ii will both be pivotal to forma basis