What are some disadvantages of using a basis whose elements are not orthogonal? (The set of vectors in a basis are linearly independent by definition.) One disadvantage is that for some vector $\vec v$, it involves more computation to find the coordinates with respect to a non-orthogonal basis. Are there any other undesirable outcomes?
I'm working on a simple model where it's more intuitive to use several physical but non-orthogonal entities as the basis.
Best Answer
I would like to mention advantage of orthogonal basis in following way: For simplicity, consider vector space to be real vector space $\mathbb{R}^n$.
We know that $\mathbb{R}^n$, considered as abstract space, has some basis, say $\{v_1,v_2,\cdots,v_n\}$. Please don't consider that this is standard basis, think that it is any basis. What can we do with this? Well! Given any $v\in \mathbb{R}^n$, there exists $a_1,a_2,\cdots,a_n\in\mathbb{R}$ such that $v=\sum_i a_iv_i$. We don't know here how to determine these $a_i$'s? We know only that there are such $a_i$'s.
However, if we have an orthonormal (orthogonal) basis then we can determine the coefficients $a_i$'s involved in the expression of $v$.
For example, if we consider the standard dot product, and if the above basis of $\mathbb{R}^n$ is orthonormal, then $a_i$ are determined by $$a_i=v{\circ}v_i.$$