Show that the given basis for S is orthogonal S span1 1 01

Show that the given basis for S is orthogonal. S = span{[1 1 0],[1 -1 6]}, s = [1 4 -9] Let S_1 = 1 [1 1 0] and S_2 = [1 -1 6]. Then S_1 middot S_2 =, so {s_1, s_2} an orthogonal basis for S. Write S as a linear combination of the basis vectors. (Give your answer in terms of S_1 and S_2.) S =

Solution

s1.s2 = 1.1 + 1.-1 +0.6 = 1-1+0 = 0

Since s1.s2= 0 , {s1,s2} is an orthogonal basis for S.

s as a linear combination of s1 and s2

c₁s₁ + c₂s₂ = s

c1 ( 1,1,0)+ c2 (1,-1,6) = (1,4,-9)

(c1+c2, c1-c2 , 0+6c2) = (1,4,-9)

(c1+c2, c1-c2 , 6c2) = (1,4,-9)

Hence,

   6c2 = -9   then     c2 = -3/2

   c1+c2 =1

   c1 - 3/2 =1

   c1 = 5/2

Hence,

   c1 = 5/2 , c2 = -3/2

Therefore, s as a linear combination of s1 and s2 is

s = (5/2)s₁ -(3/2)s₂