\fSolution : 3 Given that , T: IR -> IR be a linear transformation such that T ( x1, 2(2 ) = (x,+ 22 , 5x, +32(2) Suppose, x = ( 1, m) GIR such that T ( X ) = ( 4, 30 ) . - NOW , T ( X ) = T ( 1 ,m ) [.: x = (4 ,m ) ] = ( 1+m, 5 1+ 3m) T ( x( 1 , M12 ) = ( 2( , + 2 1 2 2 52 , + 32 1 2 ) so , T ( 1, m ) = ( 1+m, 51+3m) . . T ( x ) = ( 1 +m, 54 + 3m) From using (i ) ; we get ( 1 +m, 51+3m)= (4, 30) Itm = 4 .: ( a, b ) = ( c,d ) = ) five and ( => a = c, b = d in IR z 5 4 + 3m = 30 By the definition of To find the value of I apply the operation , two equal vectors [vs - (iv) x 3 ); we get 5 1 + 3m/ = 30 3 1 + 3m 712 ( - ) 2 1 + 0 = 18 -) 21 = 18 = ) 1 = 18 =9 2 :.1= 0 (vil From ( iv ) using ( vi ); we get g t m = 4 = ) m = 4 - 9 = - 5 m = - 5 " , m = - 5 (vii ) 50 , X = ( 1 , m ) = (9 , - 5 ) . | By ( vi ) , ( vii ) ] Hend , X = (9 , -5 )