Curiosity about complete basis

[issacnewton]

curiosity about "complete" basis

Hi

In QM books , people talk about complete basis. I was checking some linear algebra books.
Of course , we have a concept of basis in linear algebra. But these books nowhere talk
about "complete" basis. Maybe math people have some more technical term for that.
Check wikipedia article here

http://en.wikipedia.org/wiki/Basis_(linear_algebra)

if we search for the word "complete" , there is one word there, which is a hyperlink to the
following article about complete space.

http://en.wikipedia.org/wiki/Complete_space

so are physics books thinking about this when they say "complete" basis ?
in my opinion , physicists should stick to the terminology commonly used in
mathematics , so that there is no confusion when we read math books.
mathematics is the language of physics.

thanks

 

[HallsofIvy]

Being "Cauchy Complete" (your second link) has nothing to do with what basis you use so I don't believe that is meant. I am not an expert here but I suspect it just means a basis for the entire space rather than just a "partial" basis that would span a subspace.

 

[issacnewton]

hi, so physicists are being sloppy with the language ?

 

[mikeph]

Your first link is broken.

 

[Doc Al]

I am not an expert here but I suspect it just means a basis for the entire space rather than just a "partial" basis that would span a subspace.

That's how I understand it.

 

[issacnewton]

Your first link is broken.

fixed

 

[issacnewton]

would it not be better for physicists to stick to the terminology used by mathematicians ?
after reading physics books, if I want to understand the math concepts , its easier if physics community sticks to the same terminology

 

[jewbinson]

"partial basis"...?

In all topics of maths I am aware of (in particular vector spaces and other topics of algebra), the definition of basis is a set that spans the main set (e.g. linear combination of elements of the basis form the vector space V), and also all of the elements of the basis are linearly independent from one another.

A basis MUST be both linearly independent and span the vector space V.

 

[issacnewton]

hi jewbinson, that's what i was confused about. why can't physics books be content by
saying just "basis" instead of "complete basis" since math books give the definition of the basis as you said, period. when one reads the words "complete basis" ,it causes one to wonder if there is also some basis which is not complete

 

[jewbinson]

I don't know. We don't use the term "complete basis" in maths, simple as that. Sorry I can't be of more help.

 

[Claude Bile]

What is probably meant is that the basis vectors span a predefined space, as opposed to a subspace of said predefined space.

In Maths, it is nonsensical to specify a complete basis since the spaces one can define is infinite. In physics though, the number of physically meaningful spaces one can define is quite limited and thus specifying "complete" basis of a predefined space (such as R^3), whilst subject to convention, can be meaningful.

in my opinion , physicists should stick to the terminology commonly used in
mathematics , so that there is no confusion when we read math books.
mathematics is the language of physics.

I had to laugh at this. No offence to the poster mind you (who makes a valid point), its just that in my experience, physicists tend to resist mathematicians' influence fairly stubbornly.

Claude.

 

[issacnewton]

What is probably meant is that the basis vectors span a predefined space, as opposed to a subspace of said predefined space.

In Maths, it is nonsensical to specify a complete basis since the spaces one can define is infinite. In physics though, the number of physically meaningful spaces one can define is quite limited and thus specifying "complete" basis of a predefined space (such as R^3), whilst subject to convention, can be meaningful.



I had to laugh at this. No offence to the poster mind you (who makes a valid point), its just that in my experience, physicists tend to resist mathematicians' influence fairly stubbornly.

Claude.

thanks claude , makes sense about "complete" basis.

yes physicists tend to resist mathematicians' influence but at what cost ... increasingly
i find it frustrating experience to read physics books , especially when authors give
"proofs" of some theorem , which is related to physics. arguments are often hand waving.

i am currently studying Daniel Velleman's "How to prove it:A structured approach" . its really
enlightening experience. physics depts. should ask their students to take as many pure
math classes as possible. what do you think .

 

[LostConjugate]

Your right, "Complete Basis" is almost never mentioned in a LA class. It is important to define a complete set of basis vectors in QM though, so the term comes up a lot.

Math makes my head hurt. English makes it hurt more.