Question

Let T R3 R4 be the linear transformation defined by T(π1, Ο2, 73) - ( 3α1 -4 , X3, 12.x2 3.x3, 6x1-25x3, 10x2 + 10x3) (a) Determine the standard matrix representation of T (b) Find a basis for the image of T, Im(T), and determine dim(Im(T)) (c) Find a basis for the kernel of T, ker(T), and determine dim(ker(T))

Answer 1

T(4189, x) = (3x-4xg 3, xitaxg+ 3x3 6x1-x+ 5*3 lox H013) TC1,00) = ( 3,116,0) = 32+eq+683 + oily TOO,0) = (-4,2-,10) = -40,+Bef-lez Holy TOP1) = (1, 3, 5, 10) = 7.21+3 lat! Now Standard mateix is As I 9 3 AK I b) Now Reduce A to echelon form Rg er "rig 37 Rgt Rat 39 3-4 to . R3 R3-bh 1. Ryt to Ry OOO orqan sig 3 of 00- -oow of -go Do - Lo So basis of Range T dim (image T) = 2 is f (1, 2, 3), (0,1,1) Ab

c) Now by 0 Null space is solution set of Ax 20 Цю АкҶx3 = х 3х1 , ОЧXI after Reducing AX=0 to echelon form and using @ we get roa 370*7 oooo Jou 01 | 2 Too O JE .xtgy+32=0, 4+2=0*/y=-2 -2) a) Kest's basis is { fitl) dim ( Null T)=1 TT