Bilinear maps and inner product

[sphlanx]

Homework Statement

We are given a linear map f, f:R²xR²->R. f has the following properties:
1)It is linear for the changes of the first variable
2)It is linear for the changes of the second variable
3)f((3,8),(3,8))=13

We are asked to say if f is anyhow related to the normal inner product of R²

Homework Equations

The Attempt at a Solution

I know that the inner product of R2 is a bilinear map like f. If f actually WAS the linear product then it should be f((3,8),(3,8))=73. Perhaps I could say that f is the inner product, multiplied by some value? I don't really understand the question, but I have posted it exactly as stated.

 

[HallsofIvy]

You mean if it really were the inner product. A bilinear map can have any value at all on pairs of basis vectors and then is determined for all other pairs. In particular, a bilinear map on R² must be of the form f((a,b)(c,d))= k₁ac+ k₂bc+ k₃ad+ k₄bd. The "normal inner product of R²" is the specific bilinear form f((a,b),(c,d)= ac+ bd. In other words, k₁= k₄= 1 and k₂= k₃= 0.

Here, we know only that f((3,8),(3,8))= 9k₁+ 24(k₂+ k₃)+ 64k₄= 13. There are, of course, an infinite number of choices for the k's that would give that. I see know way, on the basis of a single value, to say that it is "related" to the inner product any more than any other bilinear map,