From Vector Spaces to Periodic Functions

From Vector Spaces to Periodic Functions

Vector Spaces

A fascinating result that appears in linear algebra is the fact that the set of real numbers R is a vector space over the set of rational numbers Q. This may appear surprising at first but it is easy to show that it is indeed so by checking that all eight axioms of vector spaces hold good:

1. Commutativity of vector addition:
   x+y=y+x for all x,y∈R.

2. Associativity of vector addition:
   x+(y+z)=(x+y)+z for all x,y,z∈R.

3. Existence of additive identity vector:
   We have 0∈R such that x+0=x for all x∈R.

4. Existence of additive inverse vectors:
   There exists −x∈R for all x∈R.

5. Associativity of scalar multiplication:
   a(bx)=(ab)x for all a,b∈Q and all x∈R.

6. Distributivity of scalar multiplication over vector addition:
   a(x+y)=ax+by for all a∈Q and all x,y∈R.

7. Distributivity of scalar multiplication over scalar addition:
   (a+b)x=ax+bx for all a,b∈Q and all x∈R.

8. Existence of scalar multiplicative identity:
   We have 1∈Q such that 1⋅x=x for all x∈R.

This shows that the set of real numbers R forms a vector space over the field of rational numbers Q. Another quick way to arrive at this fact is to observe that Q⊆R, that is, Q is a subfield of R. Any field is a vector space over any of its subfields, so R must be a vector space over Q.

Problem

Here is an interesting problem related to vector spaces that I came across recently:

If you want to think about this problem, this is a good time to pause and think about it. There are spoilers ahead.

Solution

The axiom of choice is equivalent to the statement that every vector space has a basis. Since the set of real numbers R is a vector space over the set of rational numbers Q, there must be a basis H⊆R such that every real number x can be written uniquely as a finite linear combination of elements of H with rational coefficients, that is, x=∑a∈Hxaa where each xa∈Q and {a∈H∣xa≠0} is finite. The set H is also known as the Hamel basis.

In the above expansion of x, we use the notation xa to denote the rational number that appears as the coefficient of the basis vector a. Therefore (x+y)a=xa+ya for all x,y∈R and all a∈H.

We know that ba=0 for distinct a,b∈H because a and b are basis vectors. Thus (x+b)a=xa+ba=xa+0=xa for all x∈R and distinct a,b∈H. This shows that a function f(x)=xa is a periodic function with period b for any a∈H and any b∈H∖{a}.

Let us define two functions: g(x)=∑a∈H∖{b}xaa,h(x)=xbb. where b∈H and x∈R. Now g(x) is a periodic function with period b for any b∈H and h(x) is a periodic function with period c for any c∈H∖{b}. Further, g(x)+h(x)=(∑a∈H∖{b}xaa)+xbb=∑a∈Hxaa=x. Thus g(x) and h(x) are two periodic functions such that their sum is the identity function.

References

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