0
$\begingroup$

I'm having some trouble with this problem that I have been working on for the past couple of hours. I was wondering if someone could please point me in the right direction.

Consider the set $S = \{A, B, C, D\}$. I need to find a basis for the vector space $F(S, \mathbb{R})$ (the set of functions that map from $S$ to $\mathbb{R}$). Then, I need help in defining an inner product on the vector space to make the basis orthonormal?

My attempt:

I believe the basis should be a set of functions, for example, $S = \{f(A) = 1, f(B) = 2, f(C) = 3, f(D) = 4\}$. I do not have much reasoning for this, though. I am just using past problems to guide me.

Any assistance is appreciated.

$\endgroup$
1
$\begingroup$

The first thing to notice is that we can map $F$ to $\mathbb{R}^4$ by mapping each $f \in F$ to its tuple of values at the "points" $A$, $B$, $C$ and $D$ i.e.

$\rho : F \rightarrow \mathbb{R}^4 : \rho(f) = (f(A), f(B), f(C), f(D))$

Note that $\rho$ is a 1-1 mapping so we can define its inverse:

$\rho^{-1} : \mathbb{R}^4 \rightarrow F : \rho^{-1}( (a,b,c,d) )= f : f(A)=a, f(B)=b, f(C)=c, f(D)=d$

We can then turn $F$ into a vector space by "pulling back" the natural vector space structure on $\mathbb{R}^4$:

$f + g = \rho^{-1}(\rho(f) + \rho(g)) = \rho^{-1}((f(A)+g(A), f(B)+g(B), f(C)+g(C), f(D)+g(D)))$

$kg = \rho^{-1}(k\rho(f)) = \rho^{-1}((kf(A), kf(B), kf(C), kf(D))) \text{ for } k \in \mathbb{R}$

and we can "pull back" the standard definition of inner product as well:

$<f,g> = \rho^{-1}(<\rho(f),\rho(g)>) = \rho^{-1}((f(A)g(A), f(B)g(B), f(C)g(C), f(D)g(D)))$

The inverse mapping $\rho^{-1}$ can then be used to map any orthonormal basis in $\mathbb{R^4}$ to an equivalent orthonormal basis in $F$. As Berci points out in their answer, a natural choice is the image of the basis $\{(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)\}$ - but that is not the only choice available.

$\endgroup$
0
$\begingroup$

What you provided is (the set of values of) a single function $S\to\Bbb R$ and not a set of functions.

For $X\in S$, let $e_X:S\to\Bbb R$ assign $1$ to $X$ and $0$ to every other elements.
Verify that $\{e_A, e_B, e_C, e_D\}$ is a basis.

Then define the inner product the usual way on coordinates w.r.t. this basis.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.