I am very confused how to go about finding an orthonormal basis using a orthogonal basis. My book says to just normalize the vectors but it doesnt further explain. When i look at answers for transforming an orthogonal basis to an orthodonal basis, i see each orthogonal vectors scaled by some crazy fraction with what is usually a squared root in the denominator. There has to be SOME trick to do this besides guessing. I just cant believe that i'm suppose to be able to guess what scalar for each orthogonal vector gives an orthodonal basis. For example what if a vector is R^10!?
1 Answer
The key is, as the book says, to normalize each of the vectors. That is, we want to replace each vector with a multiple of that vector which has length one. In other words, we want the unit vector in the same "direction".
The length of a vector $\vec x = (x_1,\dots,x_n)$ is given by $$ \|\vec x\| = \|(x_1,\dots,x_n)\| = \sqrt{x_1^2 + \cdots + x_n^2} $$ The vector $$ \frac{\vec x}{\|\vec x\|} = \left( \frac{x_1}{\sqrt{x_1^2 + \cdots + x_n^2}} ,\dots, \frac{x_n}{\sqrt{x_1^2 + \cdots + x_n^2}} \right) $$ will always have a length of $1$. That is, $\left\|\frac{\vec x}{\|\vec x\|} \right\| = 1$.
So, we can simply replace each vector $\vec v$ with $\frac{\vec v}{\|\vec v\|}$ to go from an orthogonal basis (of non-zero vectors) to an orthonormal basis.
