Inner Product on a Real Vector Space

I hope that answers your question. The inner product of u and v in your example is 97, but the notation \langle \cdot | \cdot \rangle: V\times V \rightarrow \mathbb{R}̣ just tells us that the inner product is a mapping from V×V to R, and it uses the notation < , > to represent the inner product. The specific rule for calculating the inner product is not specified in this notation.
  • #1
DUET
55
0
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?
 
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  • #2
DUET said:
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 [itex]\langle \cdot | \cdot \rangle: V\times V \rightarrow \mathbb{R}[/itex]̣.

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.
 
  • #3
DUET said:
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 [itex]|u||v|cos(\theta)[/itex] where |u| and |v| are the lengths of the vectors and [itex]\theta[/itex] is the angle between the two vectors. Those are each real numbers and so [itex]|u||v|cos(\theta)[/itex] is the product of three real numbers so is itself a real number.
 
  • #4
DeIdeal said:
Maybe this will help you: An inner product (to reals) is a map [itex]\langle \cdot | \cdot \rangle:V\times V \rightarrow \mathbb{R}[/itex]̣.

What does VxV mean?
 
  • #5
DUET said:
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.
 
  • #6
DeIdeal said:
[itex]\langle \cdot | \cdot \rangle:[/itex]̣
What does it mean? How can I read it?


Is <u, v> a Cartesian product of u and v?
 
  • #7
DUET said:
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 [itex]f:\mathbb R \rightarrow \mathbb R[/itex], 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)
 
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  • #8
DeIdeal said:
Maybe this will help you: [itex]\langle \cdot | \cdot \rangle: V\times V \rightarrow \mathbb{R}[/itex]̣.
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
[itex]\langle \cdot | \cdot \rangle: V\times V \rightarrow \mathbb{R}[/itex]̣

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

u×v→r
 
  • #9
DUET said:
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
[itex]\langle \cdot | \cdot \rangle: V\times V \rightarrow \mathbb{R}[/itex]̣

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 [itex]<,> : {\mathbb{R}}^n \times {\mathbb{R}}^n \rightarrow \mathbb R[/itex] such that for [itex]u,v \in {\mathbb{R}}^n[/itex] the pair [itex](u,v) \in {\mathbb{R}}^n \times {\mathbb{R}}^n[/itex] and [itex] (u,v)\mapsto \sum_{i=1}^n u_i v_i[/itex] which is the same as saying [itex] <u,v>=\sum_{i=1}^n u_i v_i[/itex].

In your specific example, with the regular dot product [itex] <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[/itex], 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 [itex]\sum_{i=1}^n u_i v_i\in \mathbb{R}[/itex]., as the components [itex]u_i,v_i[/itex] are all real numbers.
 
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  • #10
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.
 
  • #11
DUET said:
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 [itex]A,B\in {\mathbb{R}}^{n\times n}[/itex], [itex]\mathrm{Tr}(B^{T}A)[/itex] defines an inner product. In this case, [itex]V={\mathbb{R}}^{n\times n}[/itex].

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

Both of them are to reals.
 
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1. What is an inner product on a real vector space?

An inner product on a real vector space is a mathematical operation that takes two vectors and produces a scalar value. It is a generalization of the dot product in two-dimensional and three-dimensional space. The inner product can be defined in various ways, but it must satisfy certain properties such as linearity and symmetry.

2. How is an inner product calculated?

The calculation of an inner product depends on the definition of the inner product being used. In general, the inner product of two vectors u and v can be calculated as the sum of the products of their corresponding components. For example, in Euclidean space, the inner product of u and v is given by u1v1 + u2v2 + ... + unvn, where u1, u2, ..., un are the components of u and v1, v2, ..., vn are the components of v.

3. What is the significance of the inner product in real vector spaces?

The inner product has many applications in mathematics and physics. It is used to define the length of a vector, the angle between two vectors, and the notion of orthogonality. It also plays a crucial role in the study of linear transformations and matrices, as well as in the formulation of quantum mechanics.

4. Can the inner product be negative?

Yes, the inner product can be negative. In fact, the sign of the inner product depends on the angle between the two vectors. If the angle is acute, the inner product will be positive, and if the angle is obtuse, the inner product will be negative. If the angle is a right angle, the inner product will be zero.

5. How is the inner product related to the norm of a vector?

The inner product is used to define the norm (or length) of a vector. The norm of a vector is the square root of the inner product of the vector with itself. In other words, the norm of a vector u is given by ||u|| = √(u · u). This definition of the norm satisfies the properties of a norm, such as being non-negative and following the triangle inequality.

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