I have the following question to answer, but what I would like is a sane and easy to grasp idea of what Normalization is all about(the why, when etc) and then how to do it without it getting over one's head please.
Find an orthonormal basis of $\mathbb{R}^2$ by applying the Gram-Schmidt orthonormalization method to the vectors: $$(1; 2)^T, (2; 1)^T$$
Find a unit vector which is orthogonal to all vectors in the subspace of ${\bf R}^3$ given by: f(x; y; z)T : 3x - 2y + z = 0g