Abstract algebra vector space problem

[Acce]

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?

 

[Office_Shredder]

$$<v,v_1>=<xv_1+yv_2,v_1>=x<v_1,v_1>+y<v_2,v_1>$$
and
$$<v,v_2>=x<v_1,v_2>+y<v_2,v_2>$$

If you know all the inner products how would you solve for x and y?

You can't really do something like what you're trying, because v isn't a vector in R², and if you want to convert it to one the only way that you have is to get f^-1(v), which is what the problem is in the first place

 

[Acce]

$$<v,v_1>=<xv_1+yv_2,v_1>=x<v_1,v_1>+y<v_2,v_1>$$
and
$$<v,v_2>=x<v_1,v_2>+y<v_2,v_2>$$

If you know all the inner products how would you solve for x and y?

You can't really do something like what you're trying, because v isn't a vector in R², and if you want to convert it to one the only way that you have is to get f^-1(v), which is what the problem is in the first place

I see, I went wrong trying to think in R² too much..
As for the solution, apparently I just have to assemble those inner-products into a matrix now. Could you tell me how you came up with those inner-products? Was it just because there wasn't anything else known? Or was it a sophisticated guess based on experience? I still can't figure out, if those inner-products have somekind of meaning, or are they there just because they happen to solve this particular problem?