Are Inner Products and Dot Products Fundamentally the Same?

[dimensionless]

Is there any difference between an inner product and a dot product?

 

[Dragonfall]

A dot product is a specific inner product. An innner product is a whole class of operations which satisfy certain properties.

http://mathworld.wolfram.com/InnerProduct.html

 

[arildno]

Yes:
The dot product is an inner product, whereas "inner product" is the more general term.

EDIT:
I'm getting old. But then again, how could I ever compete with young, strong dragons swooping down on its prey?

 

[dimensionless]

What would be an example of an inner product that it not a dot product?

 

[arildno]

Well, as I'm used to it the term "dot product" is usually reserved for an operation on a finite Euclidean space; I haven't seen the term "dot product" being used for inner products defined on function spaces, for example.

 

[LeonhardEuler]

What would be an example of an inner product that it not a dot product?

Just expanding on what arildno said, one specific example is the Frobenius inner product of two matrices. It is defined by:
<A,B>=tr(B*A)
Where B* is the conjugate transpose, or adjoint of B.

 

[TD]

What would be an example of an inner product that it not a dot product?

For continuous functions from [a,b] to R, you can define an inner product <.,.> as:

\left\langle {f,g} \right\rangle = \int\limits_a^b {f\left( x \right)g\left( x \right)dx}

This is also an example on that Mathworld page.

 

[Hurkyl]

What would be an example of an inner product that it not a dot product?

(a, b) . (c, d) = ac + 2bd
(a, b) . (c, d) = (a+b)(c+d) + (a-b)(c-d)

(a) . (b) = 2ab

 

[dimensionless]

(a, b) . (c, d) = ac + 2bd
(a, b) . (c, d) = (a+b)(c+d) + (a-b)(c-d)

(a) . (b) = 2ab

Are these postulates?

 

[arildno]

No, you can show that they satisfy the PROPERTIES of the inner product:
To take the first:
1. (a,b).(a,b)=a^2+2b^2>0 unless a=b=0
2. (a,b).(c,d)=ac+2bd=ca+2db=(c,d).(a,b)
and so on with the rest of an inner product's properties.

 

[Astronuc]

I wish someone had explained that difference to me 30+ years ago, when I was first learning about vectors and multidimensional analysis. I got to university thinking that 'dot' and 'inner' product were the same, and it was terribly confusing when an inner product was introduced with a different meaning than a dot product.

 

[arildno]

The way I sort of organized the concepts to myself was like this:
General terms:
Inner/Outer products
Special terms:
Dot/cross products
Scalar/vector products

This was at least helpful for me.

 

[Dragonfall]

General terms:
Inner/Outer products

There's an outer product now?

 

[daveb]

Yup. It's commonly called the tensor or direct product.

 

[Hurkyl]

Are these postulates?

No -- they are definitions of three different inner products. The first two are on R², and the third is on R.

 

[HallsofIvy]

The theoretical "meat" of the Gram-Schmidt orthogonaliztion process is that any inner product is a dot product in some basis. Given an inner product, choose a basis and use Gram-Schmidt to derive an orthonormal basis {e₁, e₂,...,e_n}. For any vectors u,v, write u= a₁e₁+ ... , v= b₁e₁+... . Then the inner product <u,v>= a₁b₁+... + a_nb_n.

Take a look at Hurkyl's examples:
(a, b) . (c, d) = ac + 2bd
(a, b) . (c, d) = (a+b)(c+d) + (a-b)(c-d)

(a) . (b) = 2ab
Starting from the basis {(1, 0), (0,1)}, (or just (1) for the third example) use Gram-Schmidt with each of these inner products to derive an orthonormal basis and show that the inner product is the dot product in that basis.