Question

1. A Western student paid someone $250 to find elementary row operations transforming the augmented matrix (A|b) to the matrix (1 6 0 -3 010) 0 0 1 4 0 7 (RS) = 0 0 0 0 15 10 0 0 0 0 0 The Western student has no idea what this means, but you do. Assuming the field is R, describe the solution set to the system of linear equations (A[6] (equivalently, the matrix equation AX = b). Your answer should be in the form Xo + span{U1, ..., Uk} where {U1, ..., Uk} is a basis for the nullspace of A. 2. Let S = {(1, 2, 1,0), (-1,0,1,2), (0,1,2,3)} and v = (2,3,2,3), all in R4. Determine whether v E span(S) by (i) setting up the problem as a system of linear equations, (ii) applying row operations to the augmented matrix (A|6), and (iii) interpreting your result from (ii).

Answer 1

Ig Is ) 7. • o o o 1, - 3 4 o|o| о оло o o o o o | Hi, uz, n5 basie variable. na, ny free var. Кs = 5 13- 14 14 = 1 А и и + 6 х - 3x4 = 0 : =111 — — 6х2 +374) general solution — Gх, ЗҮ4 | how 2 оло тоо и 1 - 4 14 14 о on - Мо - 24 ч, + 74 Ч. keye = no + span{u,, U23 Ko= o o o 5) и = (-6 гоо ) u, = (зо -4 1 )

it vespan by s, then there exist hinang) sit ni (1,2, 1 a) + H21, 0, 1, 2) +4 (0, 1, 2, 3) - (3, 3, 2, 3). - 2-N2 = 2 anit 43=3 ni+H2 +213=2 242+ 343 = 3 (A16) بر II -1 0 121 در R₂ ~ R₂-2 en N - R3 R₃-R 10 2 2lo o - an id lo 2 33 و II R₂ R₃-R₂ 10 RGTR-R3 lo 0 1 2 1 1 -1 012 11-1 REARGR o 2 1+21) To o la pan K A 7 Rank (A16) & system is there inconsistent. is no solution (2, 22, n.) er & span (s).