1 0-5 Let A-0 4 -2and bDenote the columns of A by a,, a2, a, and let W Span(a,a2
ID: 3117056 • Letter: 1
Question
1 0-5 Let A-0 4 -2and bDenote the columns of A by a,, a2, a, and let W Span(a,a2a3) -2 82 a. Is b in fa,,a2,a^)? How many vectors are in fa,a2,a^)? b. Is b in W? How many vectors are in W? c. Show that a3 is in W. [Hint: Row operations are unnecessary] a. Is b in (a,a2.aj]? 0 O Yes How many vectors are in (a,a2,a3l? OA. One OB. Three C.Two O D. Infinitely many b. Set up the appropriate augmented matrix for determining if b is in W (Simplify your answers.) Is b in W? A. No, because the row-reduced form of the augmented matrix does not have a pivot in the rightmost column O B. No, because the row-reduced form of the augmented matrix has a pivot in the rightmost column ° C. Yes, because the row-reduced form of the augmented matrix does not have a pivot in the rightmost column. 0 D. Yes, because the row-reduced form of the augmented matrix has a pivot in the rightmost column How many vectors are in W? OA. Three OB, Infinitely many OC, Two OD. One c. The vector a3 is in W Spanfa,,a2,az because a3 can be written as a linear combination c1a1+c2a2 C3a3 where Thus, a3 is in W because a3a1 (Simplify your answers.)Explanation / Answer
a. The vectror b is not equal to any of a1,a2,a3 so that b is not in { a1,a2,a3}. There are 3 vectors in { a1,a2,a3}.Option B is the correct answer.
b. The augmented matrix required to determine whether b is in W is
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The vector b is in W as it is a linear combination of a1,a2,a3. Option C is the correct answer.
W has infinite vectors, as all linear combinations of a1,a2,a3 are in W. Option B is the correct answer.
c. The vector a3 is in W = span{ a1,a2,a3} because a3 can be written as a linear combination c1a1+c2a2+ c3a3 where c1 = c2 = 0 and c3 =1.
Thus a3 is in W because a3 = 0a1+0a2+ 1a3.
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