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University of Toronto St. George

Mathematics

MAT224H1

Sean Uppal

Fall

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MAT224H1b.doc Linear Transformations E XAMPLES AND E LEMENTARY P ROPERTIES Definition If V and W are two vector spaces, a funT :V W is called a linear transformation if it satisfies the following axioms: T1: T v +V v1)= T V +W T v1)v v1V T V T v 1 . T2: T r V v = rW T v ,r R,vV . Example a+bc a+b+c Define T : 2 M 22, andv = a+bx+cx P T2v = ( ) 1 . Show that T is linear. 2 a+b+c ab+c T 1 T v v 1)= T a a 1)+(b b 1) + c c 1) 2) . = 1 a a 1 + b b 1 c c)1( a a 1 + b b 1 + c c(1 ) ( ) 2 (a a 1 +)b b 1 + c c 1 a a 1 b b 1 + c c(1 ) ( ) 2 2 T v + T v1 = T a+bx +x )+T a1+ b1x+c 1 ) = 1 a + + + c + 1 a1+ b1 c1 a1+ b1+ c1. 2 + + c a +c 2 a1+ b1+c1 a1 b1+ c1 1 (a+a 1 +) +b 1 c+) 1 a +)1 + b+b1 + c+ 1 ) ( ) = 2 (a+a +) +b + c+) ( a+a ) (+b + )+( ) ( ) 1 1 1 1 1 1 T 2 1 ra rb rc ra rb rc T r v )T ra rbx rcx 2)= 2 ra rb rc ra rb rc . =r 1 a + + + c = T(v) 2 + + c a +c Therefore, T is linear. Example The following are linear transformations: 2 D : n P n1 where D p xn= p x n ex: D x +3x = 2x +3 . x I;Pn Pn+1 where I p n = ) ap n dy .( ) Theorem Let T :V W be a linear transformation. 1) T 0) = 0 . V W 2) T v = T v ,vV . n n 3) T a v = a T v( ). i1 i i i1 i i Page 1 of 15 www.notesolution.com MAT224H1b.doc Theorem Let T :V W and S :V W be two linear transformations. Suppose that V =span v , ,v }. If 1 n T vi= S v ii , then T = S . n n n Proof: Let v = aiviV . So T v = T aivi = aiT v i , and ( ) i=1 i 1 i=1 n n S(v)= S a v = a S v . Thus,(T ))= S v). i i i i i=1 =1 Theorem Let V and W be vector spaces, ande1, ,en} a basis oV . Given any vectow 1, ,wnW , there exits a unique linear transformatioT :V W satisfyingT e )= w ,i . In fact, the actioTis as follows: i i n n Given v = a iv iV , thenT v = aiT vi .( ) i1 i=1 Example 1 0 2 0 1 Find a linear transformatioT: P2 M 22 such thatT(1+ x =)0 0 , T x + x )= 1 0 , and 2 0 0 T 1+x )= . 0 1 2 2 (1+ x, x + x , 1+ x } is a basis Pf2 a + bx+ cx2 =c (1+ x)+c x( +x 2)+c 1+ x2) 1 2 3 a +bc c1= a c1 c3 = 0 2 2 a +b+c . (a c1 c3 + b c 1 c2 x + c c 2 c3x = 0 b c)1 c2 = 0 c2= c c c = 0 2 2 3 a b+c c3= 2 T v )=T c1(1+ x)+c 2( +x 2)+c3 1+ x2))= c1 1 + x)+c2 x + x2)+ c3T(1+ x2) 1 0 1 0 0 a+ c 1 0 + +b c 0 1 a b c 0 0 = c10 0 +c2 1 0+ c30 1 = 0 0 + 1 0 + 0 1 . 2 2 2 a +bc a +b+c T( )= 2 2 a +b+c a b+c 2 2 K ERNEL AND IMAGE OF A L INEAR T RANSFORMATION Definition Let T :V W be a linear transformation. Then: kerT = vV T v = 0 . Page 2 of 15 www.notesolution.com

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