Unique Linear Functionals

I am working on the following...

Let 𝐵 = { } be a basis for vector space 𝔙 .

Show that there is a unique linear functional on 𝔙, such that =

Also, show that the set of 𝑛 distinct linear functionals on 𝔙 obtained form 𝐵, are linearly independent.

I have no idea on what to do, could someone help me out?

Thanks

Re: Unique Linear Functionals

Quote:

Originally Posted by

**jnava** I am working on the following...

Let 𝐵 = {

} be a basis for vector space 𝔙 .

Show that there is a unique linear functional

on 𝔙, such that

=

Also, show that the set of 𝑛 distinct linear functionals on 𝔙 obtained form 𝐵, are linearly independent.

I have no idea on what to do, could someone help me out?

Thanks

The basic idea is this. Suppose that is a -space. You can completely specify a linear transformation from to any other -space by demanding that where is a basis for and the are just any vectors in (not necessarily different). How? Well, say you have made a choice about what the go to, you still haven't defined a transformation on itself. That said, if the map which takes is to be a linear transformation you must take each to . Thus, if is linear and satisfies then the function must be defined by the rule where is the UNIQUE representation of as a linear combination of the basis . Conversely, you can check that the map defined that way does, in fact, satisfy the condition of being a linear transformation with . So, in your case you have that and

For the second part, what would happen if (where I put to emphasize that it's the zero function)? What happens if you plug in for ?

Re: Unique Linear Functionals

Quote:

Originally Posted by

**Drexel28** The basic idea is this. Suppose that

is a

-space. You can completely specify a linear transformation from

to any other

-space

by demanding that

where

is a basis for

and the

are just any vectors in

(not necessarily different). How? Well, say you have made a choice about what the

go to, you still haven't defined a transformation on

itself. That said, if the map which takes

is to be a linear transformation you must take each

to

. Thus, if

is linear and satisfies

then the function must be defined by the rule

where

is the UNIQUE representation of

as a linear combination of the basis

. Conversely, you can check that the map defined that way does, in fact, satisfy the condition of being a linear transformation

with

. So, in your case you have that

and

For the second part, what would happen if

(where I put

to emphasize that it's the zero function)? What happens if you plug in

for

?

when you plug back in you will get zero, meaning orthogonality, meaning linear independence?

Re: Unique Linear Functionals

Quote:

Originally Posted by

**jnava** when you plug back in

you will get zero, meaning orthogonality, meaning linear independence?

Uh, not sure what you mean. You get that , right?

Re: Unique Linear Functionals

Quote:

Originally Posted by

**Drexel28** Uh, not sure what you mean. You get that

, right?

Yes all due to 0(v) correct? Since it will be zero, the the functionals are orthogonal implying linear dependence? This kind of math hurts my head way too much lol

Re: Unique Linear Functionals

Quote:

Originally Posted by

**jnava** Yes all

due to 0(v) correct? Since it will be zero, the the functionals are orthogonal implying linear dependence? This kind of math hurts my head way too much lol

Haha, you have proven that all the 's are zero, which shows that they are linearly independent.

Re: Unique Linear Functionals

Quote:

Originally Posted by

**Drexel28** Haha, you have proven that all the

's are zero, which shows that they are linearly independent.

Thank you for the help!