Question

Numbers 5-7

5. Solve the linear system and determine its solution space S. Is S a subspace of R”? Justify your answer. 3x1 + 6x3 + 3x4 + 9x5 = 0 X1 + 3x2 - 4x3 – 8x4 + 3x5 = 0 X1 – 6x2 + 14x3 + 19x4 + 3x5 = 0 6. Let S = {(2,0,7), (2, 4, 5), (2, -12, 13)}. If possible, write x = (2, 20, -3) as a linear combination of the vectors in S. This is equivalent to determining whether or not x E span {(2,0,7), (2, 4,5), (2, -12, 13)}. 7. Determine whether the set S = {-2t + t2,8 + +, - 2 + +, -4+ {2} spans P3.

Answer 1

13 06 3.97 indipendente fet A-1, 34 -8 31 Ti -6 14 19 3 R → Rio si 0 2 | 37 i 34 -8 3 II -6 14 19 3] R₂ R2-R R₃ R3-R, ſi 0 2 1 37 To 32-9 o lo -6 12 18 OJ R₂ R₃ + 2Rq Ti 0 ? 137 To 32- qol To ono o o 24 + 2x3 +84 + 385 = 0 3x2 +243 - 924 = 0 1643 = 0 = x, = o . 4 +Hy + 385 = 0 a 2x2 - 9 dy = 0 dy and as are free variables Case I fet dy=0 and 25 = -d then dq=0 and ay = 3 - X= (3, 0, 0, 0,-1) Case II. fet x5 = 0 and by = 2 then h=9 and a +2=0= * = -2 X= (-2,9,0,2,0). LM

** S = sparf (3,0,0,0,1), (-2,0,0,2,0]. 1. S is a subspace of Rs. (6) tet (2, 20, -3) = 64 (2,0, 75+ m2 (2,4,5) +421 2,-12, 13) fet (2, = (24+242 +243, 442-1223 , 74+542+1303) The 4+42+03 = 1 - La-343=5 74+5d2+1303 = -3 augmented matrix of the above system is. Ti 11111 0 1 -3 5 17 5 13|-3] 1 R₂ R3-IRI TL 11111 lo 1 -35 10 -2 6-10 R3 R₂ +2 R2 II [ 1111 To ļ -35 To 0 0 1 0 4+42 +d3 = 1 42-383 = 5 I let &z=0 then dq='5 and 4 =-4. .= (2, 20,-3)=-4 (2.,0, 7) + 5 (2,4,5) +0:1 2, -12,13). Hence isa linear Combinations of the vectors in S.

8 (7) fet plt) = Got qt + Qqt²+ asts and suppose that Co+Q4+ +as+*+ aşt's 1 (-2+++%+d2 (8+4°) + (3 (-+*&+dy(+4+) = 1842–404) +1-24): +ldquaq taclt? +lee tag) % 864 = 444 = Qo. - 0 -24= 9 => 49 --W2 4- da tay = Q2 => - Xat dy= Q2 +1 24. - d2 +4z = Qg. — from 0 and ③ we have 42y = 89 +49 +20.. .: Hy = 242 +94 +4 00.1 From 8&2 = Go +444 = Qo +809 +49 +20 = 44+8Q2 +2.00. de = 042 +42 + 20. from @ we have. Lg= Az-da = 93-19-92-tao .. Co + @1t+Qetºt agt = -49 (-2+++% + $(+40+29)(8++% + $1405-29-482-6)(-+*++)+7(842+49 )(4+e) thence s spans Pr. Tim X