Let u v be linearly independent vectors in a vector space V

Let u, v be linearly independent vectors in a vector space V .

(a) Find all x1, x2 R such that x1(u + v) + x2(u v) = 0.

(b) Are the vectors u + v and u v linearly independent?

Solution

(a)

x1(u+v)+x2(u-v)=0

Hence, (x1+x2)u+(x1-x2)v=0

Since u and v are linearly independent vectors.

So, x1+x2=0,x1-x2=0

Hence, x1=x2=0

(b)

Yes.

let a, b so that:

a(u+v)+b(u-v)=0

(a+b)u+(a-b)v=0

Hence, a+b=0,a-b=0

Hence, a=b=0

Hence, u+v and u-v are linearly independent