Question

Question 2. (15 pts) Let vi= [-3 0 6)". Vy=[-2 2 3". Vg= [0 - 6 3), and w=[1 14 97 (1). Determine if w is in the subspace spanned by V. V2 V3. (2). Are the vectors Vi, V2, V3 linearly dependent or independent? Justify your answer.

Answer 1

T Pet v, = [-3 o 6]" -3 o 6 V₂ = [-2 2 3]" n -2 2 3 1 > [o -6 3]" 0 E (1) of is in the subspace spanned by V., V2, 39 then w must be written written as a Linear combination V., V2, V3 of Pet W= Xvit B V2 + 8v3 -2 2 3 or 1 = -3 2-2 B 14 = 2B-68 =) p= -1-32 B = =) 2 14 +6 08 2 9- 6 at 3B +38 .. B- -1-32 1 3 = 14 +687 2 2 Take ß=1 -|-30 = 2 B-1 -) T π 14+689 = 2 -) 7+3=1 302 - 2 - |-7 6

Substituting these values in 2 we Get R.His - 6 *+ 33+37 = 6 (-1 +3(D+3(-2) - 6 + 3 - 6 = - -12 + 3 こ - 9 Hence, + L.H.S. Camat be in the subspace spanned by V, V, V.3. 7 (2) Pet av, + bvz + CV3 = 0. or a -3 tb 9 tc -6 3 6 3 o This implies -3 a-2b = 0 2b-6 c = 6a + 3b +3 = - 3 b b=3c * * gives, ac 35=} (30) or a = -20 b=30 Take C=1 a = -2 b - 3 Thus, (-2) -3 21-1100 avce Since, a, b, c 1₂ & V3 are Non-Zero. Hence vectors vi, Linearly dependent.