Question

Let TR-R be a linear transformation such that T(1, 1, 1) = (5.0,-1), TO. -1, 2) = (-5, 2, -1), and 7(1.0, 1) = (1, 0, 1). Find the indicated image 712,-1, 1) T2,-1, 1) = Need Help? Read it Talk to a Tutor Submit Answer

Answer 1

are sohn At first we ceill check that elements (11!), (0, 1, 2) and (1o, 1) lineartry independent not in IR? To check this will evaluate the oe determinant -1 2 - 1(-1-0)+1(2-0) + (OFI) E-1 +2+1 = 2 #o CS Scanned with CamScanner

non- zero Since is IR3 since 3 IR 3 3 3 IR? the determinant is the elements (11),(0, -1,2) and (1,0,1) are linearty independent dimension of and (lo,1), 10,-1)2), (1,0,1) are linearly independent in IR3 in IR3, these these elements span and hence {(1, 1, 1), (0, 1, 2), (1,083 is basis of IR Therefore any element of be written as the linear combination of the basis vectors. :: 12,-1, 1) = x (1,1,1) + 710,-1, 2)+2(1.0.1- where scalaus real Roy, z Ferom (i) we get (2,-1,1) = (x+Z ,X-7, x+2y+z) (ii) x-1=-1 vii) x+24+2 = - (iv) solring (ii), (iR) & civ) cue get are 2 7+Z = (ii) - (i) =) ztry = 3 = y = 3-z Now #tery +z =) -) 7+2 (3-2) + z = 1 -) X-Z + 6 = 1 - 5 A - Z - CS Scanned with CamScanner

(1) (v)) 2기 -3 3 )- بعدم 2. u) - (v) - ) 2급 - ) - 구 2. Noce 라상 =3 DJ=3 C 3로 2 z) - 2 Herce we can write 3 (2)-1) 1) - (1,1,1) - 글 ()-1, 2) + 골 (11) 2 I Now T(2,-11) - [ - (11) - ( , -1, 2) 4 (1,017 3 2. T(11) 2 2 T(o) -1,2) + 3 T(101) [ Since T:IR? IR is linear transformat 15 3 ) 5 2 ) ~ 1 + 2 - - 글 (5,) - (-5, 21) + 골 (1,0,1) 블 ) ( o) = (-) T(2)-1) 1) - ( -1) 호 ( 즉 클 2 (and CS Scanned with CamScanner