Simple Questions - January 11, 2019

https://math.berkeley.edu/~peyam/Math110Sp13/Handouts/Dot%20products.pdf

section 2, a proof on the generalised concept of an inner product, i have a few questions:

  • <e_i, e_j>, isn't this just the inner product between two distinct basis vectors? why would this fill an nxn Rn matrix?

  • are we assuming this e-basis is exactly an identity matrix, or at least a diagonal matrix with no entries off-diagonal?

  • if the basis is NOT diagonal, wouldn't we have waaaaayyy more terms in <x,y>, because there would be the sum of <x_i e_i, y_j, e_j> for ALL i AND all j, so instead of n terms, nxn terms. this one really confuses me. if the basis is simply a diagonal matrix, then each entry <x_i e_i, y_j, e_j> where i =/= j would simply be something times 0.

i've been so confused for a while.

/r/math Thread