What Does Completeness of Basis Vectors Mean in Three-Dimensional Space?

[PhysicsMark]

Homework Statement

If there exists no function, f(x), except zero, with the property that

$$\int_{a}^{b}{\phi_{n}(x)}f(x)w(x)dx=0$$

for all $$\phi_{n},$$ then the set {$$\phi_{n}(x)$$} is said to be complete.

Write a similar statement expressing the completeness of a set of basis vectors in three-dimensional Euclidean space. Explain in simple terms what it means for vectors.

Homework Equations

w(x) is the positive real weight function (I do not know what this means)

I'm planning on involving the inner and/or outer product here.

The Attempt at a Solution

I do not have a solid understanding of the equation above. Here is what I do know:

I know that in 3D Euclidean space the basis vectors are:

$$\hat{x}= (1,0,0)\qquad\hat{y}=(0,1,0)\qquad\hat{z}=(0,0,1)$$

I am also familiar with their inner and outer product results. I know that in 3-D it is possible to have a vector(non-zero) whose cross or dot product with 1 of the basis vectors yields 0. I know of only 1 vector(the null vector), in 3-D, whose cross or dot product with all of the basis vectors, in turn, equals 0.

I think I can answer this problem if I know what $$\phi_{n}$$ represents.

In the case of the vectors, does it represent $$\hat{x},\hat{y},{and},\hat{z}$$?

Or does it represent only 1 of the base vectors? Since I have said that I can think of a vector that will yield 0 crossed or dotted with 1 base vector, then I think $$\phi_{n}$$ would have to mean all of the basis vectors. If that is the case, am I thinking about 3 different integrals or am I thinking about 1 integral with 5 terms being multiplied?

I think it has to be 3 different integrals but I do not understand how the notation expresses that. I am familiar with the summation notation $$\hat{x}_{i}$$. Is that what we are talking about here?

Also, are we talking about the inner product, outer product or both?

One last question unrelated to this problem. I am not very confident in my Latex ability and I often preview my posts many times. Every time I do this, the original structure of the post (The problem statement, relevant Eqs, and solution attempt) get added each time I preview my post. I have to delete the repeats each post. Does anyone know a way around this?

Sorry for rambling on. Thank you for taking the time to read/respond to my questions.

 

[elibj123]

The integral is a definition of an inner product in a vector space which happens to be the space of all integrable function over [a,b]. Here $$\phi_{n}(x)$$ and f(x) play the role of vectors, while w(x) is some intrinsinc function of the inner product, which is not a vector (the function itself might be a vector, but the role w(x) plays is not as a vector in this space).
You can recognize this is an inner product, because that takes two vectors and returns a scalar.

This alone helps you understand that when talking about cartesian spaces, we will be talking about dot products, and not cross or outer products.

Now, the $$\phi$$'s don't have to be your "standard" basis (what's the standard basis for functions?), but rather they are some set of infinite functions, which may or may not span the entire space.

Since you know what a dot product means, you can have a sense of what this inner product means. And perhaps you can take it on from here.

 

[PhysicsMark]

Thank you for the reply

You can recognize this is an inner product, because that takes two vectors and returns a scalar.

This makes sense to me.

Now, the $$\phi$$'s don't have to be your "standard" basis (what's the standard basis for functions?), but rather they are some set of infinite functions, which may or may not span the entire space.

If the phi's are an infinite set of functions, does that mean in my problem it represents every possible vector in 3-D?

If that is the case, why does the problem word it as "the completeness of a set of basis vectors..."? I read the set of basis vectors in 3-D to be the 3 basis vectors I defined earlier.

Is the correct answer:

If there exists no vector, V, except the null vector, with the property that

$${\hat{x}_i}\cdot{V}=0$$

for all $$\hat{x}_i$$, then the set {$$\hat{x}_i$$} is said to be complete.

That seems correct to me, but I am not entirely sure.

 

[vela]

If the phi's are an infinite set of functions, does that mean in my problem it represents every possible vector in 3-D?

If that is the case, why does the problem word it as "the completeness of a set of basis vectors..."? I read the set of basis vectors in 3-D to be the 3 basis vectors I defined earlier.

No. Don't forget you're dealing with two different vector spaces here. One is the space of functions, which has infinite dimension, which, in turn, means a basis must consist of an infinite set of functions, and the other one is R³, which has only dimension 3. The problem is just asking you to draw an analogy between the phis and a basis of R³, not every possible vector in R³.

Is the correct answer:

If there exists no vector, V, except the null vector, with the property that

$${\hat{x}_i}\cdot{V}=0$$

for all $$\hat{x}_i$$, then the set {$$\hat{x}_i$$} is said to be complete.

That seems correct to me, but I am not entirely sure.

You got it. Now you just need to explain what completeness means in terms of the vectors in R³.

 

[PhysicsMark]

You got it. Now you just need to explain what completeness means in terms of the vectors in R³.

Thank you for reading and verifying. Based off this I believe that completeness means that any vector in $$R^3$$ can be expressed/written as a linear combination of the 3 basis vectors.

 

[vela]

Yup, that's it.