Question

1 O -7 5 Let A= 0 2 -2 and b= - 1 Denote the columns of A by ay, az, az, and let W= = Span{a,,a,,az}. -34 2 -5 1 a. Is b in {a4, az, az}? How many vectors are in {aq,a2, az}? b. Is b in W? How many vectors are in W? c. Show that az is in W. (Hint: Row operations are unnecessary.] a. Is b in {aq, az, az}? No Yes How many vectors are in {aq, az, az}? O A. One B. Infinitely many C. Two D. Three b. Set up the appropriate augmented matrix for determining if b is in W.

Answer 1

© Herce A> 0-7 0 2 -2 1-3 4 2 5 and ba 15 Then az A2? 2 2 4 Az 2 and Wo span da, ag, ass afw; wo gaitez aytgag fore a.cz,eger] @ The set Las, aq, az} does not contain b, it only contain the vectors a, A2, A₃- Thus, 3 veetorn are in fa, a2, AzJ.i.e, option @ is right 6 Fore determing it bis in w We set the augmented matrix as [A: b] 0 2 5 -7. -2 2 -3 ५ -5 1 5 Pstopy 2. -2 4 -19 Oo R3-282 2 Rank A=3 NOW- Rank [A:b] Thus, the reeduced forem of the matrcine has a pivot in the right most column. This implies b in Wo spandai, az, az]. augmented

Thes, oprien @ is correct. o since we spanda, as, As} slow; Gaur ez a te az fore any G, l2, and e, e2 ls are in RJ. arbitrary, then w contain reight Here Waspandai, az, az}. infinitely many vectors. Thus, option @ is so clearely az is in W os az zoa, to do tl. az