Inner Product on a Real Vector Space

[DUET]

Let V be a real vector space. Suppose to each pair of vectors u,v ε v there is assigned a real number, denoted by <u, v>. This function is called a inner product on V if it satisfies some axioms.

1. What does refers by "this function"? Is it "<u, v>"? If it is then How we can call it's a function?

2. What does "there is assigned a real number" suggest? Could someone please explain it to me with examples?

 

[DeIdeal]

Let V be a real vector space. Suppose to each pair of vectors u,v ε v there is assigned a real number, denoted by <u, v>. This function is called a inner product on V if it satisfies some axioms.

1. What does refers by "this function"? Is it "<u, v>"? If it is then How we can call it's a function?

2. What does "there is assigned a real number" suggest? Could someone please explain it to me with examples?

Maybe this will help you: An inner product (to reals) is a map $\langle \cdot | \cdot \rangle: V\times V \rightarrow \mathbb{R}$̣.

1. Functions (maps, mappings) are relations between sets of "inputs" and "outputs", the "input" being the two vectors of V, the "output" being the result of the inner product. Why do you think it can't be called a function?

2. "There is assigned a real number" means that the map is to reals (In this case. It doesn't have to be), ie. that when you take an inner product of two vectors, the result is a real number.

 

[HallsofIvy]

Let V be a real vector space. Suppose to each pair of vectors u,v ε v there is assigned a real number, denoted by <u, v>. This function is called a inner product on V if it satisfies some axioms.

1. What does refers by "this function"? Is it "<u, v>"? If it is then How we can call it's a function?

Are you saying you do not know what a "function" is? A function "from A to B" is a collection of pairs, (x, y), such that x is from A, y is from B, and no two different pairs have the same first member. That is, if (x, y) and (x, z) are in the collection, y must equal z. The dot product assigns, to every pair of vectors a specific number. A is the set of pairs of vectors and B is the set of real numbers.

2. What does "there is assigned a real number" suggest? Could someone please explain it to me with examples?

If u= (1, 0, 1) and b= (2, 1, 1) then <u, v>= 1(2)+ 0(1)+ 1(1)= 3. The number "3" is assigned to that pair of vectors.

Or, assign to vectors u and v the real number $|u||v|cos(\theta)$ where |u| and |v| are the lengths of the vectors and $\theta$ is the angle between the two vectors. Those are each real numbers and so $|u||v|cos(\theta)$ is the product of three real numbers so is itself a real number.

 

[DUET]

Maybe this will help you: An inner product (to reals) is a map $\langle \cdot | \cdot \rangle:V\times V \rightarrow \mathbb{R}$̣.

What does VxV mean?

 

[verty]

What does VxV mean?

This is set theory, it is the Cartesian product of V with V, so the set of ordered pairs of elements of V.

 

[DUET]

$\langle \cdot | \cdot \rangle:$̣

What does it mean? How can I read it?


Is <u, v> a Cartesian product of u and v?

 

[DeIdeal]

What does it mean? How can I read it?Is <u, v> a Cartesian product of u and v?

(I'm not a native speaker, but) I'd read it as "An inner product map" (and the followup would be "from V times V to R"). I'm just declaring the function there, just like you'd do $f:\mathbb R \rightarrow \mathbb R$, for example.

<u,v> is not the cartesian product of u and v, it's the inner product of u and v. V×V is the cartesian product of two "copies" of the whole vector space V. It's not a single element, but, like verty said, it's another vector space that contains all (ordered) pairs of two vectors of V. You choose an element of V×V, say (u,v), to take the inner product <u,v> with. (Also I find it funny that I accidentally wrote the inner product with the |-bar. Never mind that)

 

[DUET]

Maybe this will help you: $\langle \cdot | \cdot \rangle: V\times V \rightarrow \mathbb{R}$̣.

Let,
u, v ε V

u=( 2, 4, 5, 7) and V = (1, 3, 4, 9)

Now what is the inner product of u and v?

I would request to discuss it by illustrating the following
$\langle \cdot | \cdot \rangle: V\times V \rightarrow \mathbb{R}$̣

Is it mean inner product of the above two vector is -

u×v→r

 

[DeIdeal]

Let,
u, v ε V

u=( 2, 4, 5, 7) and V = (1, 3, 4, 9)

Now what is the inner product of u and v?

I would request to discuss it by illustrating the following
$\langle \cdot | \cdot \rangle: V\times V \rightarrow \mathbb{R}$̣

Is it mean inner product of the above two vector is -

u×v→r

No, it's to say that not u×v (that rarely means anything outside R³ afaik), but the pair (u,v) ∈ V×V maps to ( ↦) some specific real number r ∈ R. In addition to knowing from which space and to where the product is from, you must also know what inner product it is. That is, a rule according to which you actually calculate the product.

IF you consider, as an example, the regular "dot product" in Rⁿ, the inner product is $<,> : {\mathbb{R}}^n \times {\mathbb{R}}^n \rightarrow \mathbb R$ such that for $u,v \in {\mathbb{R}}^n$ the pair $(u,v) \in {\mathbb{R}}^n \times {\mathbb{R}}^n$ and $(u,v)\mapsto \sum_{i=1}^n u_i v_i$ which is the same as saying $<u,v>=\sum_{i=1}^n u_i v_i$.

In your specific example, with the regular dot product $<u,v>=\sum_{i=1}^n u_i v_i=2\cdot 1+4\cdot 3+5\cdot 4+7\cdot 9=2+12+20+63=97$, but notice that there are various different inner products in Rⁿ, and you can't say anything about the value of <u,v> unless you specify the inner product. For Rⁿ, it usually is the dot product, but it can be something else as well.

Notice that $\sum_{i=1}^n u_i v_i\in \mathbb{R}$., as the components $u_i,v_i$ are all real numbers.

 

[DUET]

Yes! it's now pretty clear. If there is any more question then I will come back again.

One more:
As I know all the dot products are inner products but all the inner products are not dot product. Could you please give me an example of inner product which is not dot product.

 

[DeIdeal]

Yes! it's now pretty clear. If there is any more question then I will come back again.

One more:
As I know all the dot products are inner products but all the inner products are not dot product. Could you please give me an example of inner product which is not dot product.

Well, if what I wrote above is how you'd define a dot product, then sure. I'll give two fairly common examples while I'm at it.

For real-valued square matrices $A,B\in {\mathbb{R}}^{n\times n}$, $\mathrm{Tr}(B^{T}A)$ defines an inner product. In this case, $V={\mathbb{R}}^{n\times n}$.

For real-valued continuous functions f,g on interval [a,b] the integral $\intop_a^b f(x)g(x) \mathrm{d}x$ defines an inner product. In this case, $V=C([a,b])$ (that being the "space of continuous real-valued functions on the interval [a,b]" like I said).

Both of them are to reals.