Question

; Let at be a linear transformation as follows : T{x1,x2,x3,x4,x5} = {{x1-x3+2x2x5},{x2-x3+2x5},{x1+x2-2x3+x4+2x5},{2x2-2x3+x4+2x5}]

a.) find the standard matrix representation A of T

b.) find the basis of Col(A)

c.) find a basis of Null(A)

d.) is T 1-1? Is T onto?

Answer 1

T(x1, 22, 23, 24, 25) = {(x,- x₂ + 2x2 + xg), CX2 X3 +285), (x, +82-2X2 + x4 +2xg), (2x2 - 2x3 +24+225)} find the standard matrix representation A ofT T(1,0,0,0,0) = (toto) f(0,1,0,0,oftor, I, 22 - 1(1,0,0,0,0) + (0,1,0,0,0) t160,0,1,0,0) + 0(0,0,0,1,0) +0(0,0,0,0,1) T(0,1,0,0,0) (2, 1, t, 2) = 211,0,0,0,0) +110,1,0,0,0)+1(0,0,1,0,0) +210,0,0,1,0) +010,0,0,0,1) T(0,0,1,0,0) = (-1,-1, -2,-2) = -1(1,0,0,0,0,0] -110,1,0,0,0) - 2(0,0,1,0,0)2(0,0,0,1,0) to(0,0,0,0,1) T(0,0,0,1,0) =(0,0, 1, 1) oll,0,0,0,0) + (0,1,0,0,0) +110,0,1,0,0) + 110,0,0,1,0) +010,0,0,0,1) T(0,0,0,0,1)= (1, 2, 2, 2) - L11,0,0,0,0) + 2(0,1,0,0,0) +2(0,0,1,0,0) +2(0,0,0,1,0) + 0(0,0,0,0,1)

A -1 OL -1 o 2 Tooror T T 2 O -2 L - 2 t оо 2 2 o find the basis of collas convert A into echelone from R3 R₃-RI T 2 1 - 1 0 1 -1 0 2 1 1 -2 1 2 -1 -I Сооо 2 o - R3 R3TR2 ㅗ 2 - 1 0 1 ㅗ -1 02 -2 3 2 - 2 12 Ra R4-2R2 I 2 L -1 ot -102 + 3 O T -2 -1 basis of colla) = {D) NHẠdo

a basix of Null (A) (co find Parc = $(x,x2, 11317*83 | TCX, 42,43,44) = (0,0,0,0)) (x, 7X2-2X ₂ +X 4+28), (282-283 +24+25) >(0,0,0,0)] x-X₂+2x2+x=0 X-Xz+ 2xg x, tx₂-2X₂ +X4+2x₂=0 222-2X₂ + x4 +22=0 1 1 x 2 -1 a -1 o 0 2 1 1 -2 L 1 2 0 2. -2 1 2 оо les B convert it into echelone from (harto) 1 I, 2 -1 I -LO 2 -2 1 3 I-2 12 23 0

x+222-x₂+x=0 X2-23 + 2x5=0 - 2x 3+x4+x=0 xq-2x=0 X4=28 - 2X3 + 2Xg+3x70 x, tts -Sxg+x=0 x + (455) XS=0 xs 2 12-+235-0 x 2 Nullspace- XER - sxs 2x5 xs let x=1 basis of Null(A) = NINGINN

becaus Kealt) is not (d) T is not one-one equal to jero. ie. Hull space have Monjeno tem etes Vector but I is onto because Rankit) = 4 -ie. dim (COLLA)) = 4