How to evaluate Nonorthogonal basis?

[marshall.L]

hi all~
How to evaluate the performance of a set of nonorthogonal basis?
Like one in Hibert space which is most likely to be a nonorthongal set.
Does it have any advantage compared with orthogonal ones in any aspects?

i don't even know every to get started

 

[HallsofIvy]

I am not sure what you mean by "evaluate". "Orthonormal" bases have the nice property that the coefficient of the basis vector $\vec{e}_i$ in the expansion of $\vec{v}$ is just the dot product: $\vec{e}_i\cdot\vec{v}$.

Other than that, there is nothing special about orthogonal bases.

 

[marshall.L]

I am not sure what you mean by "evaluate". "Orthonormal" bases have the nice property that the coefficient of the basis vector $\vec{e}_i$ in the expansion of $\vec{v}$ is just the dot product: $\vec{e}_i\cdot\vec{v}$.

Other than that, there is nothing special about orthogonal bases.

thx:)

i mean whether there is any kind of measurement which can be used to evaluate any aspect of a set of basis vectors.
i.e. we can use reconstruction error to evaluate the descriptive ability of a set of basis vectors.(The only way i know)

i havnt learned much on this aspect and i have searched on wikipedia for a long time with no progress.

i don't know whether i have made my question clear.
sry for my poor eng.