Question

Let A = 10-o] 0 2 -3 and b = - 4 4 3 . Denote the columns of A by a, a, az, and let W = Span{a.a.az). a. Is b in aa.az)? How many vectors are in a .a .a? b. Is bin W? How many vectors are in W? c. Show that a is in W. (Hint: Row operations are unnecessary.] a. Is b in {a .az.az)? No Yes O How many vectors are in aaa? A. Three OB Infinitely many OC. Two OD. One b. Set up the appropriate augmented matrix for determining if b is in W. (Simplify your answers.) Is b in W? A. Yes, because the row-reduced form of the augmented matrix does not have a pivot in the rightmost column. OB Yes, because the row-reduced form of the augmented matrix has a pivot in the rightmost column. O C. No, because the row-reduced form of the augmented matrix does not have a pivot in the rightmost column. OD. No, because the row-reduced form of the augmented matrix has a pivot in the rightmost column. How many vectors are in W? O A Three OB. Two Ос. One D. Infinitely many c. The vector a, is in W-Span(a, a, a) because a, can be written as a linear combination ca, + C a + cza, where C1, C2, and C are scalars. Thus, ay is in W because a = a1 + a2 +0a3. (Simplify your answers.)

Answer 1

Columns of A are a, a, az. So, ay> [t] - [] - 13 (as I is not equals to al az, az So, 64 {a,, 92, 93}. there are there 3 rectos in 29, 392, 93}. (b) 1: 1 (6+12) - 4(12) o 2 - 31 T-4 4 3 1 &0. . a,, az, az is lineasly independent vectors So, ha az, az is basis of R3 sopSp. So, Span {a,, az 193} =W=IR3 So any vector in R² can be expressed as S linear combination of as, an, a3. & E span {an, an, az}=W. These are infinility many vectors in w. o a, 2 la, + 0.0₂ +0. 93. a, & span {a, , A2, A3} 2w. 2) atw Corred,