Linear Algebra 1 an 1 and 0 6 Consider the three vectors in

(Linear Algebra)

1 an 1 and 0 6. Consider the three vectors in part (c) of problem 1. Find a single vector v such that its inner product with every one of the vectors in 1(c) vanishes. What is the dot product of this v with an arbitrary linear combination of the vectors in 1(c)? What is the sum of components of an arbitrary linear combination of the vectors in 1(c)?

Solution

Let v=(a,b,c)

So, v.(1,-1,0)=0

So, a=b

Similarly taking dot product with other two vectors gives

b=c=a

So, v=a(1,1,1)

Set a=1 and we get one such vector. Though for all a in R we have v satisfying given conditions

Denote given three vectors by :x1,x2 and x3 respectively

So arbitrary linearly combination is

(rx1+sx2+tx3)

v.(rx1+sx2+tx3)=rv.x1+sv.x2+tv.x3=0

(rx1+sx2+tx3)=(r+t,-r+s,-s-t)