Explain briefly why if p n then u1 up is a basis of Rn Giv

Explain briefly why if p = n then {u_1, ..., u_p} is a basis of R^n. Give an example of a basis B = (v, w) of R^2 and scalars a, b R such that ||av + bw||^2 notequalto a^2 + b^2. Prove that if p = n, then ||x||^2 = (x middot u_1)^2 + ... + (u middot u_p)^2. Prove (when p

Solution

a> Since {u1,u2, ...un} is an orthonormal set of vectors(p=n), they are linearly independent. Also the dimension of Rⁿ =n= number of vectors contained in the set {u1,u2, ...un}

Therefore, it will be a basis of Rⁿ