Question

Consider the linear transformation T : R2 + R2 defined as T(21,12)=(0,21 – 12). Find the standard matrix for T: a ab sin(a) 8 f E д 0 0 1 What is the dimension of ker(T)? Is T one-to-one? no 47 Enter one: yes no Write the standard matrix for HoT, where H is the reflection of R2 about the z-axis. a ab sin(a) f 12 II 8 R ат

Answer 1

xet pol 7 TOIR? (R² be a cineas transformation defined by T(A4, R2) = (0, x-xe e, (1.0), e2= (0.D} be the Stasidard basis for 1R? Then TCE) o T(10) col) 10.(1,0) +7.601 2 TCes) . T (ol) (02-1) 0 (10) + (-). (0.1) 2 2 Standard matrine fos T is [TIB = [: (x, y) EIR? and exet (x,y) t kere Then T(**)=(0,0) or cox y) = (0,0) => a=y. (x,y) = (x,x) 2:2 (1:1). Therefore basis for kert is { (1.13

dim (ker (1)) = 1, WA Kes (T) 7o so I is not Since one-one , 1R2 H 1R2 is the reflection about the xsaris Then it (ay) = (2,-Y) 2 2 Now HOT (ny) 2 H (T(24) Hco, n-1) C0,-(4-1)). HOT (e) HOT (10) Co,-1) 0.(10) + 6-1):(0,1) HOT () - HOT C0-1) (0,-(0-1)) (0.1) OC1,0) + 1. (0.1). and 9 [Hot 0 'B -1 -