Question

R2 defined as Consider the linear transformation T: R2 T(21,22)=(0,21 – 22) Find the standard matrix for T: a ab sin (a) f 8 ат What is the dimensi of ker(T)? Is T one-to-one? Enter one: yes no Write the standard matrix for HoT, where H is the reflection of R2 about the 3-axis. a sin(a) f 22 8 R a E är (Alt + A)

Answer 1

Solution Given that T:1R2>R2 be a linear transformation defined by T(2,22) = (0, x-7) The standard basis of IR2 is Bipz ={(1,2), (0,1)} Now T(1,0) = (0,1-0 → T(60) = (0,1) > T(10) = O(1,0) + 1(0,1) (1) and T(0,1) = (0, 0-1 → T(31) = (0-1) → T(0,0) = 0 (10) -4(0,1) T(0) O (1,0) + 160 (0,1) = (2) Therefore, from (1) and (2), we get the matrix of linear trans formation Twith respect to the standard basis Bipz = {(10),(0,1)} is the

T(10) T(0,1) (10) o 0 (0,1) -1 1 Therefore the standard matrix of T is Q 0 0 BIR? [T] BIR² = 1 -1 . { since T(x1,2) =(0,21-23) Finding Kar (7) Kar(t) = {(x,x){IR T(x,x2) = (0,0)} Kar(t) = (x,x)EIR? : (0,*-*:) = (0,0)] 7 kar (t) = {(2x2) E1R2: *-*=0} → Kar (1) = f (x,x) EIR2: X, =} Kar(t) = |(x2,-2): «ER} Kar (1) = f(50): fxz (11) : WEIR] ㅋ

=> B Kors (T → Kar(t) = sband (103 Basis of Kar (T) is {(1,1)}. basis of Kar (1), Bear (0) = {(1,03 only Since contains one vector Then dim (kar (7)) = 1 Therefore dim (kar (t)) = Answer) Since dim (Kar(t)) = 10 a Then by the theorem? T is not Theorem! Let T: 1R IR" be linean transformation Then T is an one to one maß if and only is dim (Kar(t)=0 an one to one linear trans formation. Therefore T is not an one to one linear trans formation (Answer