I remember my professor saying there are certain advantages to using an orthogonal basis. One is that it's easy to determine the coordinates of a given vector. For example, we are familiar with the canonical {$[0,1], [1,0]$} basis of $\mathbb{R^2}$. But doesn't the orthogonality (and the inner product) depend on the coordinate system?
To illustrate, let $V_1 = [0,1]$ and $V_2 = [1,0]$ in canonical coordinate system. Then, let's use the basis vectors {$[0,1],[0.5,0.5]$} and use coordinates relative to this new basis. Then, relative to the new basis, $V_1$ has coordinates $[1,0]$ and $V_2$ has coordinates $[2,-1]$. So the inner product $V_1 \cdot V_2 = 1*2 + 0*-1 = 1$ is not longer zero, and thus $V_1$ and $V_2$ are no longer orthogonal in the new coordinate system.
Questions:
1. Is my understanding correct? Is it true that the inner product depend on the basis vectors and the coordinate system?
2. If yes, then are there any deficiencies to using {$[0,1],[0.5,0.5]$} as the basis for $\mathbb{R^2}$?
Best Answer
The inner product does not in fact depend on a particular coordinate system. The error you are making is in the statement:
"Then, relative to the new basis, V1 has coordinates [1,0] and V2 has coordinates [2,−1]. So the inner product V1⋅V2=1∗2+0∗−1=1 "
Summing up the product of coordinates of two vectors gives the inner product only if the coordinates are taken with respect to some orthonormal basis. Since V1,V2 do not form an orthonormal basis, the system of coordinates induced by them cannot be used in the usual formula for the inner product.