Let V be an inner product space of dimension n and let u and

Let V be an inner product space of dimension n, and let u and v be vectors in V . Furthermore, suppose that v can be written as a linear combination of the vectors v1 , v2 , . . . , vn , all of which lie in V : v = c1v1 + c2v2 + · · · + cnvn . Use the properties of the inner product to write the inner product as a linear combination of the inner products for i = 1, . . . , n.

Solution

Let u = a₁u₁ + a₂ u₂ +... +a_n u_n. Since v = c₁v₁+c₂v₂+...+c_nv_n , hence u.v = a₁c₁ u₁v₁ +a₂c₂ u₂v₂ +...+ a_nc_n u_n v_n = a₁ c₁( u₁ . v₁ ) + a₂c₂(u₂.v₂) +...+a_nc_n(u_n.v_n).