1. ## Othonormal Basis

Let B= { $w_1, w_2, ... w_k$} be an othonormal basis for a subspace W and let v be any vector in W, where v = $\lambda_1w_1+\lambda_2w_2+...+\lambda_kw_k$.
Show that $|v|=\sqrt{\lambda_1^2+\lambda_2^2+...+\lambda_k^2}$

2. ## |V|

$|v|=\sqrt{v\cdot v}=\sqrt{\lambda_1 w_1 \lambda_1 w_1+...+\lambda_n w_n \lambda_n w_n}=\sqrt{\lambda_1^2+...+\lambda_n^2}$

This clear if you understand what orthonormal means.

An orthonormal basis means span, linear independence, multiplication is the kronecker delta (1 if same index 0 otherwise), each has magnitude 1.

3. So if I have a orthogonal basis ${(1, -2, 0, 2)^T (16, 13, -9, 5 )^T (2, 9, 21, 8)^T}$. How would I show the orthonormal basis?

4. If they are orthogonal already you just divide each one individually by their magnitude, this will make them unit length, ie normalized. This will not affect their orthogonality and they will become orthonormal.