- #1

Acce

- 2

- 0

## Homework Statement

We have a vector space (V, R, +, *) (R being Real numbers, sorry I couldn't get latex work..) with basis V = span( v1,v2). We also have bijection f: R² -> V, such as f(x,y) = x*v1+y*v2.

Assume you have inner-product ( . , . ): V x V -> R. ( you can use it abstractly and assume the axioms hold ). Now we can assemble the inverse map of the given bijection f, such as f⁻¹: V -> R². With the help of inner-product, form a matrix equation to find the coefficient pair f⁻¹(v) in R² for any arbitrary v in V.

## The Attempt at a Solution

I have tried to assemble the matrix, but my problem is, I don't understand how to use the inner-product as an abstract object. I don't see how it helps me to know that i get a real number from the inner-product, since I fail to see what that real number expresses here. As far as I have understood, the inner-product can be defined the way the modeler/mathematician as long as the axioms still hold. Could someone give me any pointers in this? Should I just read definition of inner-products more carefully?

...

edit: I figured out something, which seems to be close to it, but it apparently is satisfactory only if basis is an orthogonal basis.

I figured that since we have inner-product, we also have norm, so we get

A=[[norm(v1) 0];[0 norm(v2)]],

X=[x y],

B=v,

when equation is A*X=B. I'm now struggling to get it work for all basis. Could someone atleast tell me have I got it right so far?

Last edited: