Use the dot product to show that if S v1 v2 vn is an ortho

Use the dot product to show that if S = {v_1, v_2, ....., v-n} is an orthogonal set of nonzero vectors in R^n then S is a linearly independent subset of R^n.

Solution

Let, a1,...,an so that

a1v1+...+anvn=0

Taking dot product with vi gives

aivi.vi=0

HEnce, ai=0 since vi is non zero vector and vi.vi=||vi||^2

Hence, ai=0 for all 1<=i<=n

Hence the set S is linearly independent