Question

V1 = 1 , V2= -1 , U3 = , 04 = 1 , 05 = 6 -3 0 | 2 Let S CR5 be defined by S = span(01, 02, 03, 04, 05). A. Find a basis for S. What is the dimension of S? B. For each of the vectors 01, 02, 03, 04, 05 which is not in the basis, express that vector as linear combination of the basis vectors. C. Consider the vectors W1 = 14, W2 = -1 Determine if each of these vectors is in S. If the vector is in S, write it as a linear combination of the basis vectors for S you found in the first part.

Answer 1

1 ) here 9 เมโส - S3 - • ( - o - weite all these in materia form: C9 2 1 1 2 1 | 2 - 1 - 2 ( 1 , A (Bes) -3 - 3 0 - 2 3 1-2 1 6 - 1 Abely 8, 9 R4 A s | 2 44 2 | - 2 - 3 0 3 1 2 2 1 1 2 1.2 1 61 | J

Apply R2 R₂-2R, 1 R3 R3 + 2 R, 1 Ru"Ru-2R1, RST Rstar, An rii la 11 lo -3 -3 4- To -3 -1 -2 - loo - 30 Lo q 8 -9 31 Apply R3 - R₂-DR2 , Rs Rs +322. Ao l'11111 10 -3 -3 4 Joo 2 06 0 Toot 30 Lo O -1 3 od apply Rs 7&s-Ry: - Anii 1 1 1 1 1 1o-3 - 342 Oo + 3 O Loo 0 0 0 apply: R4 - Rut 1/2 Rg: - To-3 -3 47 o o 2-6 0 10 0 0 0 0 Loo 0 0 0 1 which is row-echelon form of A. Rank (A) =3.

(A) Lince S= span {n902, 03, Qur 53. - dim (S)=3. . (B) Lo out of 5 vector, 2 vectos are not in besis. {01 102 104} is basis of s. [ U 0.19 03 are linegoly independent ?

Uz= (-2,-5,-3,0-3) = a (2,2,1,1,2) +6 (21/14, 211) tcllil,1,1,1). => 24+26+c=-2 – cis 2a-+c= -5 - ciis a-6+c = -3 – cils a +26-6=0 — (iv) 2a+b+c= -3 - (V) from here, we get a=-2, b=1,c=0 » E82-57-370,-3) = -2 (21241,112) + (214,71221). Os= (-2,7161-7, V = 9 (212111,2) +6(2-1,41201) 1 tc (lelelet, I). 5) 2a +267 C=-2 29 -b +c=7 a-b+c=6 a +26-65-7 2a +b+c=1 from here we get aal, b=-3, c=2. .: (-217,6 -7,1)= (21211112) -3 (2 +242211) + 2 (111114,1).

(0) W,- ( 9261 42/28) .a (242,1,1,2) +6C24 Hotell h tc Cieloletel). 20+26+6=9 za -670= 6 a-btc=4 atzb-c=1 22+b+c=8 here we geti- a=2 16=1,2=3 : (926,4,118) = 2 (212111112) 41 (217,4,2,1) 7311,1,117,1). Wq = (3,714,212) = Q(2,2,1,1,2) + b(21-4, 4, 2, 1) + C (lellited) - 22+267 C=3 za ob+c=4 a -b+c=rl a +26-(=2 2a +b+c= 2. here, we get> a=-27 , b = 58, c=1. (3,714,212) 5 (2121111,2) +ş (214441210) + Cle11117,1). ;