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MATH-M303 Section 1.9 Notes

by: Kathryn Brinser

MATH-M303 Section 1.9 Notes MATH-M 303

Marketplace > Indiana University > Mathematics > MATH-M 303 > MATH M303 Section 1 9 Notes
Kathryn Brinser
GPA 4.0

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About this Document

Covers matrix representation of linear maps and the relations between them.
Linear Algebra for Undergraduate
Keenan Kidwell
Class Notes
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This 2 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 3 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 1.9 Notes- The Matrix of a Linear Map 9-26-16  Every linear map T:R → R m is a matrix map of form T (x)=Ax for a unique m×n matrix A th  Standard basis vectors- ei has a 1 in i spot and 0’s everywhere else o For n≥1 :  e = 1,0,0,… ,0 ) 1  e2= (0,1,0,…,0 )  e = (0,0,0,…,1 ) n o Columns of n×n idnntitm matrix o Any linear map T:R → R completely determined by its outputs on ei o Basis vectors span Rn - every vector x ϵR n is linear combination of them  x= ( 1x ,2,x n) ¿ ( ,0,0,…,0 + ) ( ,0,…,0 +…+ ),0,0(…,x ) 1 2 n ¿x 1 1x e2+2+x e n n  Coefficients in linear combination of ei , which give x , are just entries of x  Ex. n=2 (3,4 = (3,0)+(0,4) ¿3 1,0 +4 (0,1) ¿3e 14e 2  Ex. n=3 (3,4,5)= (3,0,0)+(0,4,0)+ 0,0,5 ) ¿3 1,0,0 +4 0,1,0 +5 (0,0,1) ¿3e +4e +5e 1 22 33  Ex. Let T:R →R be linear map with T ( 1 (5,−7,2 ) and T ( 2 (−3,8,0 ) . T x ,x Find T (3,4)) and (( 1 2)) . o T (3,4 )=T 3( +1e 2) ¿3T e(1)T e (2) ¿3 5,−7,2 +4 (−3,8,0 ) ¿(15,−21,6 + −12,32,0 ) ¿(3,11,6 ) o T ((,1 =2)) e (x1 1 2 2) ¿x 1 ( 1x T 2 ( 2 ¿x 15,−7,2 +x 2−3,8,0 ) 5 −3 x1 ¿ −7 8 columns are outputs of T on e1,e2 [ 2 0 [x2  Works because A x=x a1+1 a 2x2e +1 1 2 2 o Discovered T is matrix map and columns of representing matrix were outputs of T on basis vectors  Theorem- Let T:R → R m be a linear map; then m×n matrix m A= [ e(1) ( 2 e (n)] where T (i)R is the unique matrix such that T x =Ax for all x ϵR n o Given x= ( ,x ,…,x ) , compute T (x)=T ( e +x e +…+x e ) : 1 2 n 1 1 2 2 n n ¿x1T (1)x 2 e( 2+x T en ( n x 1 ¿[T( 1 ( )2T e ( n] x2 ⋮ [xn] ¿A x by definition of A and product A x o For linear map T , A is standard matrix of T , sometimes denoted [T]  Sometimes say T is represented by its standard matrix because T(x)=Ax  Not a linear map, just represents/implements it 3 3 o Ex.Find the standard matrix of T:R →R given by T (x)=6x .  T( 1=6e 1 , T( 2=6e 2 , T (3)6e 3 6 0 0  Standard matrix A= 0 6 0 [0] 0 6 3 2 o Ex.Compute the standard matrix of T:R →R given by T ((1x ,2 =3) (x1,x 2x 2 3) . Is the vector b= (3,2) in the range of T ?  T( 1=T (1,0,0)) ¿(1−0,0−0 ) ¿(1,0)  T( 2=T (0,1,0)) ¿(0−1,1−0 ) ¿(−1,1)  T( 3=T 0,0,1) ¿(0−0,0−1 ) ¿(0,−1 )  A= 1 −1 0 [0 1 −1 ]  b being in range of T→ T x =b for some x ; row reduce Ax=b  [ | ]= 1 −1 0 3 already in EF [0 1 −1 2 ]  System is consistent, and b is in range ¿Span {1,0),−1,1 ),0,−1 )}


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