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# MATH-M303 Section 1.8 Notes MATH-M 303

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This 5 page Class Notes was uploaded by Kathryn Brinser on Thursday September 29, 2016. The Class Notes belongs to MATH-M 303 at Indiana University taught by Keenan Kidwell in Summer 2016. Since its upload, it has received 4 views. For similar materials see Linear Algebra for Undergraduate in Mathematics at Indiana University.

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Date Created: 09/29/16

M303 Section 1.8 Notes- Introduction to Linear Maps/Transformations 9-19-16 n If A is m×n matrix, them for any vector xϵR , mulmiplication by A produces new vector A x ϵR ; if we regard vectors in R as inputs on which A acts by multiplication to give output iR m , and we arrive at notion of a function Function/map n m - rule which assigns unique output m to each n T:R → R T(x)ϵR input xϵR n m R R goes from domain to target o Image of under - output vector x T T(x) n o Range of T - set of all images of vectors{ (x):xϵ R} ; everything that gets hit by T Any matrix gives matrix map n m given by T (x=Ax m×n A T:R → R o Because of relation to linear systems, we can answer many questions about matrix maps with knowledge from prior sections o If A= [ 1 2a n] , then T(x)=Ax=x a1+1 a 1…2x a n n ; range of T is set of linear combinations of columns of A , which is Span {1,a2,…,a n} a1+1 a 1…2x a n n 1 −3 Ex. Let A= 3 5 , u= 2,−1 ) , b= 3,2,−5 ), c= (3,2,) [ ] −1 7 o Let T:R →R 3 be map given by T (x=Ax o (a) Find T u)=¿ image of under : u T By definition of T , T u =Au 1 −3 ¿ 2 [ ] 5 [−1 −1 7 2 1 −−3 ( ) ( ) ¿ 2( )−5 [ 2(−1)−7 2+3 ¿ 6−5 [−2−7 5 ¿ 1 [−9 2 o (b) Find an xϵR whose image under T is b (ie. T (x=b ): T (x=Ax=b 1 −3 Solve matrix equation: x1= 3 [ ] 5 x2 [] −1 7 −5 3 1 −3 3 1 0 2 [A∨b ] 3 5 2 0 1 −1 REF [−1 7 −5 [ ] 2 0 3 −1 Unique solutionx=( ), 2 2 1 −3 3 A x= 3 5 2 −1] 7 []1 2 3 1 2 ( )2(−3) 3 1 ¿ 2(3− 2(5) 3 1 [ (−1)− 7) 2 2 3 3 2 2 9 5 ¿ 2− 2 []3 7 2 −2 3 ¿[]2 correct −5 o (c) Isx=(3,2,−) unique? Determining ifT(x=b has a unique solution, which is equivalent to whether A x=b has a unique solution Yes, because the REF oA|b] has no free variables o (d) Determine ic=(3,2,) is in range oT : Range of T=¿ set of all outputs from vectoRs (ie. { x):xϵ R} ) Finding ifc is an output/if there existxϵRn such that T(x=c↔ Ax=c has a solution Row-reduce [Ac] : 1 −3 3 1 −3 3 [A|c]= 3 5 2 0 2 −1 EF −1] 7 5 |[0] 0 10 R 3 indicates A x=c inconsistent, so T(x=c has no solution, and c not in range ofT o Parts (c) and (d) answer existence and uniqueness questions for systems translated to maps 1 0 0 Ex. Let A= 0 1 0 ; map T:R →R 3 defined by T(x)=Ax [ ] 0 0 0 o Map is called projection ontox1x2 plane; fixes points i1x2 plane 1 0 0 x 1 x1 o For x=(x ,x ,x) , T( )=Ax= 0 1 0 x = x 1 2 3 [0 0 0 []2 []2 x3 0 o Graphical representation: T (5,0,)) 5,0,0) T (0,2,−1)=(0,2,) 2 o T ∘T=T =T Ex. Let J= 0 −1 and T:R →R 2 given by T x =J x . Describe this [1] 0 geometrically. o T (1)T (1,)) ¿ J e 1 ¿ 0 −1 1 [1] 0 [0 1( )−1( ) ¿[ ] 1( )−0( ) ¿ 0 =¿ first column ofJ [1 90° rotation o T (2)T (0,)) ¿Je 2 ¿ 0 −1 0 [1] 0 [1 ¿ 0 ( )1 (1 ) [ 0( )+1( ) ] −1 ¿ =¿ second column of J [ 0 90° rotation All matrix maps have 2 special properties according to theorem from 1.4; if A is m×n matrix and u ,vϵ R ,cϵR , then: o A u+v )=Au+Av and A cu =c (Au ) o T u+v )=T (u+T v) and T cu =cT u ) Linear map- map T:R → R m linear if it satisfiTsu+v )=T (u)+T v) and T(cu =cT u) for u,vϵR n and cϵR o Commutes with addition and scalar multiplication o Derivatives do this- they can be considered linear maps in a way o From here on, only working with linear maps o Ex.Nonlinear map sin x:R→R sin(π+3π =sin4π ¿0 ¿sinπ+sin3π this input works; must check infinitely many, technically π π sin(2)+ 2 =sinπ ¿0 ≠sinπ +sinπ =1+1=2 2 2 ∴ by counterexample, sin x not linear o Only functions f :R→R which are linear have form fx =ax for some fixed scalar a Properties of Linear Maps o Sends 0ϵR n to 0ϵR m , ie. T ( )0 n m T ( nT 0 (n n) T(0 )=T 0( ) 0 ( ) n n n T ( nT 0 (n) 0(+( n ( ) n)−T ( n 0 =T 0 because T 0 ϵR m m ( n (n) o T cu+d v =cT (u+dT v) ¿c T ( )c T u( )+c T u +d( )v +d ( ) +…+d ( ) ( ) 1 1 2 2 n n 1 1 2 2 n n To show linearity, must verify this for all possible inputs simultaneously 2 2 2 Ex. Show that T:R →R , T (x =4x is linear; let u,vϵR and cϵR be arbitrary and use definition of T along with vector operations to get result T (u+v =4 u+v ) ¿4u+4 v ¿T u +T (v) T (cu )=4 (cu ) ¿ 4c )u ¿ c4 )u ¿c (4u ) ¿cT u ) ?T is linear Ex. Show that T:R →R ,T x ,x = x ,x ,x +x is linear; let ( 1 2) ( 1 2 1 2) u= ( 1u 2)= v ,( 1R 2) 2 and cϵR T (+v =T)u +((,1 +v1 2 2) ¿ ( 1v 1u +2 ,u2+v1+u 1v 2 2) ¿ ( 1v 1u +2 ,u2+u1+v 2v 1 2) ¿ ( 1u 2u 1u +2) (v1,v 2v 1 2) ¿T u +T (v) T ( )=T c u(,( 1 2)) ¿T c( ,c1 2) ¿ cu ,cu ,cu +cu ( 1 2 1 2) ¿c ( 1u ,2 +1 2) ¿cT u )

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