Abstract Linear Algebra: Dual Basis

In summary, a linear functional y can be defined on C^3 such that y((1,1,1))=y((1,1,-1))=0 by setting y(x)=1*a+(-1)*b+0*c for x=(a,b,c). This satisfies the conditions and is a simple, concrete solution.
  • #1
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Homework Statement


Define a non-zero linear functional y on C^2 such that if x1=(1,1,1) and x2=(1,1,-1), then [x1,y]=[x2,y]=0.


Homework Equations


N/A


The Attempt at a Solution


Le X = {x1,x2,...,xn} be a basis in C3 whose first m elements are in M (and form a basis in M). Let X' be the dual basis in C3'. Let N be the subspace of V' spanned by ym+1, ..., yn.
Let's assume that y is any element in N.
1) y is in V'
2) y is a linear combination of the basis vectors y1, ..., yn
=> y = [tex]\Sigma[/tex]j=1n njyj
Since by assumption y is in N we have for every i=1,...,m
[xi,y] =0

P.S: I am new to the abstract linear algebra world.
 
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  • #2
Apparently you mean a linear functional on C^3. What is C? Is it the complex numbers? And I think you are taking this 'abstract' thing a little far. You just want a linear functional y such that y((1,1,1))=y((1,1,-1))=0. Try thinking of it as a 'not abstract' problem. You want to write down a concrete linear functional that maps x1 and x2 to zero.
 
  • #3
Ok, so let x=([tex]\xi[/tex]1,[tex]\xi[/tex]2,[tex]\xi[/tex]3) (where [tex]\xi[/tex]1=[tex]\xi[/tex]2) and let y be the functional such that y = [tex]\xi[/tex]1+[tex]\xi[/tex]2+0*[tex]\xi[/tex]3
So for xi (i from 0 to n) y(xi) would equal 0.
 
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  • #4
Fringhe said:
Ok, so let x=([tex]\xi[/tex]1,[tex]\xi[/tex]2,[tex]\xi[/tex]3) (where [tex]\xi[/tex]1=[tex]\xi[/tex]2) and let y be the functional such that y = [tex]\xi[/tex]1+[tex]\xi[/tex]2+0*[tex]\xi[/tex]3
So for xi (i from 0 to n) y(xi) would equal 0.

i from 0 to n? There's only x1 and x2. How about if x=(a,b,c) define y(x)=1*a+(-1)*b+0*c? Then y(x1)=0 and y(x2)=0. That's really all you need. Your notation [x1,y] has got to mean y(x1), right?
 

What is abstract linear algebra?

Abstract linear algebra is a branch of mathematics that studies vector spaces and linear transformations without relying on a specific coordinate system or basis. It uses abstract concepts and notation to describe and analyze mathematical structures, making it applicable to a wide range of fields including engineering, physics, and computer science.

What is a dual basis?

A dual basis is a set of vectors that form a basis for the dual space of a given vector space. It is a set of linear functionals that can be used to uniquely express any vector in the original space. The dual basis is often referred to as the "dual of the basis" and is important in understanding the duality between vector spaces and their dual spaces.

How are dual bases related to linear independence?

In a vector space, a set of vectors is considered linearly independent if none of the vectors can be written as a linear combination of the others. Similarly, in the dual space, a set of functionals is linearly independent if none of the functionals can be written as a linear combination of the others. The concept of dual bases allows us to extend this notion of linear independence to the dual space.

What is the importance of dual bases in abstract linear algebra?

Dual bases play a crucial role in abstract linear algebra as they allow us to define isomorphisms between a vector space and its dual space. This allows us to translate geometric concepts and operations in the original space to algebraic concepts and operations in the dual space, and vice versa. Dual bases also allow us to study properties of vector spaces and their duals simultaneously, providing a more complete understanding of these structures.

How are dual bases used in practical applications?

Dual bases have practical applications in fields such as signal processing, optimization, and quantum mechanics. In signal processing, they are used to analyze and manipulate signals in the frequency domain. In optimization, they help to solve problems involving linear functionals and constraints. In quantum mechanics, they are used to analyze the properties of quantum states and their measurements.

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