Dual spaces-Existence of linear functional

In summary, the conversation discusses the existence of a non-zero linear functional f on a finite dimensional vector field V over F, given that there is a scalar c and a vector v in V such that T(v) = cv. The discussion also involves defining f for other vectors, and it is shown that f cannot be defined for all vectors in V.
  • #1

Homework Statement


Let V be a finite dimensional vector field over F. Let T:V→V
Let c be a scalar and suppose there is v in V such that T(v)=cv, then show there exists a non-zero linear functional f on V such that Tt(f)=cf.
Tt denotes T transpose.

Homework Equations


Tt(f)=f°T. Ev(f)=f(v).
Let V*=HomF(V,F)
V*=(V*)*
E:V→V** i.e. E(v)=Ev
Theorem: E is an isomorphism iff V is finite dimensional


The Attempt at a Solution


Ev(Tt(f))=f°T(v)=cf(v).
But now I need to show there is f such that Tt(f)=cf, not just for specific v that satisfies T(v)=cv.
 
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  • #2
How would you define f?? Obviously you will want to have f(v)=v. How would you define it for other vectors?? Maybe set f(w)=0 for some other vectors??
 
  • #3
f is just a linear transformation in V*=HomF(V,F), so you cannot have f(v)=v, f:V→F.
 
  • #5
We cannot define f(kv)=1 and f(w)=0 for w≠v, since if a is a multiple of v and b is a multiple of v, a+b is a multiple of v, so 1=f(a+b)≠f(a)+f(b)=2. So f won't be a linear transformation.
 
  • #6
neomasterc said:
We cannot define f(kv)=1 and f(w)=0 for w≠v, since if a is a multiple of v and b is a multiple of v, a+b is a multiple of v, so 1=f(a+b)≠f(a)+f(b)=2. So f won't be a linear transformation.

Indeed we can't. Can you solve that?? Don't put ALL the vectors equal to 0, but only some. Do you see it??
 

1. What is a dual space?

A dual space is the set of all linear functionals on a vector space. It is denoted by V* and is also known as the algebraic dual space.

2. What is the relationship between a vector space and its dual space?

The dual space of a vector space V is a vector space itself, and its dimension is equal to the dimension of V. The elements of the dual space are linear functionals, which are linear maps from V to the underlying field. This means that every element of the dual space can be evaluated on a vector from V and will result in a scalar value.

3. How do you prove the existence of linear functionals on a vector space?

To prove the existence of linear functionals on a vector space V, we need to show that there exists at least one linear functional on V. This can be done by defining a linear functional, verifying that it satisfies the axioms of linearity, and showing that it is unique.

4. Can every vector space have a dual space?

No, not every vector space has a dual space. Only finite-dimensional vector spaces over a field have a dual space. Infinite-dimensional vector spaces may have a generalized dual space, but it is not equivalent to the dual space of a finite-dimensional vector space.

5. How is the dual space related to the concept of dual basis?

A dual basis is a set of linear functionals on a vector space that are linearly independent and span the dual space. In other words, a dual basis is a basis for the dual space. This means that every element in the dual space can be uniquely expressed as a linear combination of the dual basis elements.

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