Question

6. Check if T is linear from the definition: T(X1, X2, X3) = (*******0,5x2 – x3, 4x2) Find the matrix that describes T and determine: Is Tonto? Is T one to one? Explain your answers. 7. Use row reduction on the [AU] matrix to find A. if it exists. Try to use as few row reductions as possible to determine if it exists or not. 3 -1 2 A= 1 5 -3 -2 4

Answer 1

SOLUTION 8 be if T IS said to linear for Tovew T (U+V) = T (U+ T (V) + T (ou) = CTV) VU,VEV, CEF o Now T (X, X2,23) ( * , +3 x + 22 +23 $ 52, -23, 4x2) Let u= (x1, X2, X₂) V= ( 97, 92, 93) then I (U+V) = T (2,4-4, X+92, X3 HY3) El C2,+9) + (+ 9) + (23+92), 5(7,49,)-(73+y), 4 (23+9)) 42, +997) x1 + x2 + x3 +4, +4₂ +43 95% -2₂+SY, -43, 4X₂ +44 3. 1) ( 21, ***??, 527-73,4x2) + (9,****, 59,-43, 443) T (X, X2, X3) + T (9,42,43) T(U) + TV) (2) T (CU)= T (CX1, C2, C13) ( C3, 4cxa+C17 ,507-50X3, 4cx2) clip , 52,-23, 4x2) = c T CXC, 72,2;) x1+x2+xz 3 = CTCU) Both condition satisfied. Hence T is linear.

The given mear transformation can be written in mabix form. as + T (X1, X₂, X3 ) = 5x -xz 4x2 [g xt 5 0 X₂ + X3 24 Hence mabix that deeribes T is 1/ Y3 Y3 -1 T is one-oue said to be iff T(U)=0 u=0 ( *,**{%YS, 5*-*3, 421) 540,0,0) x2=0 450,-8320 4 24+23 20 x=0 X250 Hence. => - Cap, X2,337 = 40,0,0) T is one-one, Hence

T is onto کاند T (O)=y let e, + X2 + 23 skyras, 4+) = (4, 9, Y) x2 = 5x-x3 = 4 x+xz + 43 9 -9, 3 x1+X3 = 34,-43 9 - 3 - 634 = 9z+34, -43 7 x = the 43 24 S) Got sx - % S2 6 + su - S93 24 64 7 x3 şy, 6 S43 29 Hence we T is outo. can find as femetion of ", 2, 93. e x2, x3

7 AE 3 1 - 2 s 2 -3 we Know (A:P] = [2:41] - o 3 1 - 2 - 1 s T 2 -3 | 6o I R, Rp 3 n 1/3 1/3-2/3 S 4 2-3 l lo R2 ~ R2+RI R3 R3 -2R, n im 13 -2/3 14/15 10/3 13 0 -11/ 7/ 1-2 o - R2 R2X5/16 Y Yz u 25/24 13 5/40 5/16 o -43 0 -143 o R3 R3 + 4 R2 /3 - 2/3 [ 2/24 443 48 Y3 S/48 5/16 o -41 SS 144

R3 R3 + 1/R2 (а -2/3 У У 6 3/h 6 s) 37/g -1/6 + 1 R39 ерз 37 1 % -23 s/ Уа о б Г 3/is Ује -14 ч %9 (37 Р.. - P. - 4 1 -23 1 Why 4 - 7 –74ч 14ч (31 RIRI + 2/3 R3 16 1/27 Г. о то oo ) 14 /-ч -1/4ч 177ч /ч 16/ -/3 7 Ф)37 R, R, -1/2 R2 о 17/ny 5/74 / 1/7ч п/7ч - 7h V4 ч -S/37 oo | *(37

Hence Inverse of A 17/74 5/74 7/37 9/74 7/74 -5/37 Aya -7/74 "1/4 8137