# Abstract algebra vector space problem

• Acce
In summary, the conversation discusses how to use the inner-product as an abstract object to solve a matrix equation involving a bijection and a vector space with a basis. The solution involves using the inner-products to form a matrix and solve for the coefficients x and y. The conversation also addresses potential challenges in solving the problem and clarifies the purpose of the inner-products in this specific context.
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?

Last edited:
$$<v,v_1>=<xv_1+yv_2,v_1>=x<v_1,v_1>+y<v_2,v_1>$$
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

Office_Shredder said:
$$<v,v_1>=<xv_1+yv_2,v_1>=x<v_1,v_1>+y<v_2,v_1>$$
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?

## What is abstract algebra?

Abstract algebra is a branch of mathematics that deals with the study of algebraic structures, such as groups, rings, and fields. It focuses on the properties and relationships between mathematical objects rather than specific numerical calculations.

## What is a vector space?

A vector space is a set of mathematical objects, called vectors, that can be added together and multiplied by numbers. It is a fundamental concept in linear algebra and is used to represent geometric quantities and transformations.

## What is a vector space problem?

A vector space problem is a mathematical question that involves applying the properties and operations of vector spaces to solve a given problem. It may involve finding linear combinations, determining if a set of vectors is linearly independent, or finding a basis for a vector space.

## What are the key properties of a vector space?

The key properties of a vector space include closure under addition and scalar multiplication, associativity and commutativity of addition, existence of a zero vector and additive inverses, and distributivity of scalar multiplication over addition.

## How is abstract algebra used in real life?

Abstract algebra has many practical applications, including in computer science, physics, engineering, and cryptography. It is used to model and solve problems involving symmetry, patterns, and structures, and has also been used to develop coding and encryption algorithms for secure communication.

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