What is the Dual Basis for Linear Algebra?

In summary, The conversation is about proving that the set {f1, f2, f3} is a basis for V*, and finding a basis for V for which {f1, f2, f3} is the dual basis. This is done by using the Gauss Jordan method to show linear independence and applying the Replacement Theorem. It is also mentioned that finding the basis vectors for V involves solving a system of equations.
  • #1
pezola
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0
[SOLVED] Linear Algebra Dual Basis

Let V= R3 and define f1, f2, f3 in V* as follows:
f1 = x -2y
f2 = x + y +z
f3 = y -3z

part (a): prove that {f1, f2, f3} is a basis for V*

I did this by using the gauss jordan method and showing that {f1, f2, f3} is linearly independent. Now because dim(V) is finite, I know that dim(V) = dim(V*). Because the set {f1, f2, f3} has exactly three vectors and the dimension of V* is three, by a corollary to the the Replacement Theorem, {f1, f2, f3} is a basis for V*

part (b) Find a basis for V for which {f1,f2,f3} is the dual basis.

I know that a for a dual basis, fi(xj) = [tex]\delta[/tex]ij , but I can’t find the x1,x2,x3 for which this works. Any suggestions?
 
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  • #2
Each basis vector in V that satisfies fi(ej) = [tex]\delta[/tex] ij defines a system of 3 equations in 3 unknowns. Each of these systems gives you one of the basis vectors.
 

FAQ: What is the Dual Basis for Linear Algebra?

What is the concept of dual basis in linear algebra?

The dual basis in linear algebra refers to a set of vectors that are used to define linear functionals. These functionals are linear maps that take a vector as input and produce a scalar as output. The dual basis is important because it allows us to relate the geometric and algebraic aspects of vector spaces.

How is the dual basis related to the standard basis?

The dual basis is related to the standard basis in that it is a basis for the dual space, which is the space of all linear functionals on a given vector space. The elements of the dual basis are defined by their action on the elements of the standard basis. This means that the dual basis vectors are uniquely determined by the standard basis vectors.

What is the significance of the dual basis in linear algebra applications?

The dual basis has many applications in linear algebra, particularly in areas such as optimization, differential geometry, and quantum mechanics. In optimization, the dual basis is used to define the dual problem, which is a fundamental concept in convex optimization. In differential geometry, the dual basis is used to define the tangent space of a manifold. In quantum mechanics, the dual basis is used to represent states and observables in a vector space.

How do you find the dual basis for a given vector space?

To find the dual basis for a given vector space, you first need to determine the standard basis for that space. Then, for each vector in the standard basis, you can define a linear functional that maps that vector to 1 and all other vectors to 0. These functionals will form the dual basis for the given vector space.

Can the dual basis be used for non-Euclidean vector spaces?

Yes, the concept of dual basis can be extended to non-Euclidean vector spaces, such as those with a non-standard inner product or a non-standard norm. In these cases, the dual basis vectors will be defined by their action on the elements of the non-standard basis. However, the dual basis will still have the same properties and applications as in Euclidean spaces.

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