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Fundamental Subspaces E-Book   Chapter   Section Orthonormal Basis

If you go back to the Euclidean n-space chapter where we first introduced the concept of vectors you’ll notice that we also introduced something called a dot product.  However, in this chapter, where we’re dealing with the general vector space, we have yet to introduce anything even remotely like the dot product.  It is now time to do that.  However, just as this chapter is about vector spaces in general, we are going to introduce a more general idea and it will turn out that a dot product will fit into this more general idea.  Here is the definition of this more general idea.

 

Definition 1  Suppose u, v, and w are all vectors in a vector space V and c is any scalar.  An inner product on the vector space V is a function that associates with each pair of vectors in V, say u and v, a real number denoted by  that satisfies the following axioms.

(a)  

(b)  

(c)  

(d)  and  if and only if  

 

A vector space along with an inner product is called an inner product space.

 

Note that we are assuming here that the scalars are real numbers in this definition.  In fact we probably should have been using the terms “real vector space” and “real inner product space” in this definition to make it clear.  If we were to allow the scalars to be complex numbers (i.e. dealing with a complex vector space) the axioms would change slightly.

 

Also, in the rest of this section if we say that V is an inner product space we are implicitly assuming that it is a vector space and that some inner product has been defined on it.  If we do not explicitly give the inner product then the exact inner product that we are using is not important.  It will only be important in these cases that there has been an inner product defined on the vector space.

 

Example 1  The Euclidean inner product as defined in the Euclidean n-space section is an inner product.

 

For reference purposes here is the Euclidean inner product.  Given two vectors in ,  and , the Euclidean inner product is defined to be,

                                              

 

By Theorem 2 from the Euclidean n-space section we can see that this does in fact satisfy all the axioms of the definition.  Therefore,  is an inner product space.

 

Here are some more examples of inner products.

 

Example 2  Suppose that  and  are two vectors in  and that , , … ,  are positive real numbers (called weights) then the weighted Euclidean inner product is defined to be,

                                             

 

It is fairly simple to show that this is in fact an inner product.  All we need to do is show that it satisfies all the axioms from Definition 1.

 

So, suppose that u, v, and a are all vectors in  and that c is a scalar.

 

First note that because we know that real numbers commute with multiplication we have,

              

So, the first axiom is satisfied.

 

To show the second axiom is satisfied we just need to run through the definition as follows,

              

and the second axiom is satisfied.

 

Here’s the work for the third axiom.

                                        

 

Finally, for the fourth axiom there are two things we need to check.  Here’s the first,

                                             

Note that this is greater than or equal to zero because the weights , , … ,  are positive numbers.  If we hadn’t made that assumption there would be no way to guarantee that this would be positive.

 

Now suppose that . Because each of the terms above is greater than or equal to zero the only way this can be zero is if each of the terms is zero itself.  Again, however, the weights are positive numbers and so this means that

                                           

We therefore must have  if . 

 

Likewise if   then by plugging in we can see that we must also have  and so the fourth axiom is also satisfied.

 

Example 3  Suppose that  and  are two matrices in .  An inner product on  can be defined as,

                                                            

where  is the trace of the matrix C.

 

We will leave it to you to verify that this is in fact an inner product.  This is not difficult once you show (you can do a direct computation to show this) that

                                      

 

This formula is very similar to the Euclidean inner product formula and so showing that this is an inner product will be almost identical to showing that the Euclidean inner product is an inner product.  There are differences, but for the most part it is pretty much the same.

 

The next two examples require that you’ve had Calculus and so if you haven’t had Calculus you can skip these examples.  Both of these however are very important inner products in some areas of mathematics, although we’re not going to be looking at them much here because of the Calculus requirement.

 

Example 4  Suppose that  and  are two continuous functions on the interval .  In other words, they are in the vector space .  An inner product on  can be defined as,

                                                      

 

Provided you remember your Calculus, showing this is an inner product is fairly simple.  Suppose that f, g, and h are continuous functions in  and that c is any scalar.

 

Here is the work showing the first axiom is satisfied.

                                 

 

The second axiom is just as simple,

                         

 

Here’s the third axiom.

                            

 

Finally, the fourth axiom.  This is the only one that really requires something that you may not remember from a Calculus class.  The previous examples all used properties of integrals that you should remember.

 

First, we’ll start with the following,

                                          

Now, recall that if you integrate a continuous function that is greater than or equal to zero then the integral must also be greater than or equal to zero.  Hence,

                                                                  

 

Next, if  then clearly we’ll have .  Likewise, if we have  then we must also have .