Bilinear maps and inner product

In summary, we are given a linear map f, f:R2xR2->R, which is linear for the changes of both variables. We are asked to determine if f is related to the normal inner product of R2. However, based on the given information, it is not possible to determine this as there are infinitely many choices for the values of k in the bilinear map that would satisfy the given condition.
  • #1
sphlanx
11
0

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.
 
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  • #2
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,
 

Related to Bilinear maps and inner product

1. What is the difference between bilinear maps and inner product?

Bilinear maps and inner product are both mathematical concepts used in linear algebra. However, they have different definitions and properties. Bilinear maps are functions that take two vector inputs and produce a scalar output, while inner product is a function that takes two vector inputs and produces a scalar output, but with additional conditions such as symmetry and positive definiteness.

2. How are bilinear maps and inner product used in real-world applications?

Bilinear maps and inner product have numerous applications in various fields such as physics, engineering, and machine learning. Bilinear maps are commonly used in geometric transformations and image processing, while inner product is used in optimization problems and data analysis.

3. What is the significance of the inner product in vector spaces?

Inner product is an important concept in vector spaces because it allows us to define notions of angle and length in a vector space. It also provides a way to measure the similarity between two vectors and can be used to define orthogonal and orthonormal bases.

4. Can bilinear maps and inner product be generalized to other mathematical structures?

Yes, both concepts can be extended to other mathematical structures. For example, bilinear maps can be generalized to multilinear maps, which take more than two vector inputs. Inner product can also be extended to complex vector spaces, where it is known as a Hermitian form.

5. Are there any limitations or drawbacks to using bilinear maps and inner product?

One limitation of bilinear maps is that they do not capture all the information about a matrix, as they only take two vector inputs. Inner product, on the other hand, may not always exist for certain vector spaces, such as those with infinite dimensions. Additionally, the definition of inner product can vary depending on the field in which it is used.

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