Bases and other strange things

[Stephen P. King]

Hi Folks,

    Lizr's resent post got me thinking again about the concept of a basis and reading the wiki article brought up a question.

http://en.wikipedia.org/wiki/Basis_%28linear_algebra%29

"In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system" (as long as the basis is given a definite order)."

    The reference to that phrase that I have highlighted was unavailable, so I ask the resident scholars here for any comment on it.

-- 
Onward!

Stephen

"Nature, to be commanded, must be obeyed." 
~ Francis Bacon

[Russell Standish]

The definition is a somewhat wordy, but essentially technically
correct, form of the standard definition of a basis in Linear Algebra.

What is your question, exactly?

Cheers

On Tue, May 22, 2012 at 09:09:07AM -0400, Stephen P. King wrote:
> Hi Folks,
>
> Lizr's resent post got me thinking again about the concept of a
> basis and reading the wiki article brought up a question.
>
> http://en.wikipedia.org/wiki/Basis_%28linear_algebra%29
>

> "In linear algebra <http://en.wikipedia.org/wiki/Linear_algebra>, a
> *basis* is a set of linearly independent
> <http://en.wikipedia.org/wiki/Linear_independence> vectors
> <http://en.wikipedia.org/wiki/Vector_space> that, in a linear
> combination <http://en.wikipedia.org/wiki/Linear_combination>, can

> represent every vector in a given vector space

> <http://en.wikipedia.org/wiki/Vector_space> or free module
> <http://en.wikipedia.org/wiki/Free_module>, or, more simply put,
> which define a "coordinate system" /_*(as long as the basis is given
> a definite order*_/)."

>
> The reference to that phrase that I have highlighted was
> unavailable, so I ask the resident scholars here for any comment on
> it.
>
> --
> Onward!
>
> Stephen
>
> "Nature, to be commanded, must be obeyed."
> ~ Francis Bacon
>

> --
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--

----------------------------------------------------------------------------
Prof Russell Standish Phone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Professor of Mathematics hpc...@hpcoders.com.au
University of New South Wales http://www.hpcoders.com.au
----------------------------------------------------------------------------

[Stephen P. King]

On 5/23/2012 1:03 AM, Russell Standish wrote:
> The definition is a somewhat wordy, but essentially technically
> correct, form of the standard definition of a basis in Linear Algebra.
>
> What is your question, exactly?

Hi Russell,

Could you elaborate on the dependence of the basis being given in a
definite order?

[meekerdb]

On 5/23/2012 4:53 AM, Stephen P. King wrote:
> On 5/23/2012 1:03 AM, Russell Standish wrote:
>> The definition is a somewhat wordy, but essentially technically
>> correct, form of the standard definition of a basis in Linear Algebra.
>>
>> What is your question, exactly?
> Hi Russell,
>
> Could you elaborate on the dependence of the basis being given in a definite order?

I don't think the order of the basis elements has any significance except notationally
when a general element is expressed as an n-tuple in terms of the basis.

Brent

[Russell Standish]

Yes - that was my take on the Wikipedia article too - namely that a
coordinate system is an n-tuple or real (or complex) numbers
describing a vector space element.

In other words, a coordinate system is a mapping from V (the vector
space in question) to R^n (or C^n). For such a mapping to be unique,
we need an ordered basis supplied.

Cheers