Talk:Basis function

definition

This page should give a definition of a basis function first, bolding the key word. Fresheneesz 05:45, 23 March 2006 (UTC)[reply]

Square integrable

There is a problem about sine and cosine not being square integrable but being used as basis vectors for a space of vectors that are square integrable. If I tried to fix it, the language might not be right. David R. Ingham 01:49, 9 September 2006 (UTC)[reply]

a comment

this page is sloppily written and redundant. it omits essential mathematical details and claims to be mathematics. the topic it purports to discuss is covered in much better fashion in Hamel basis, Hilbert space, and probably other pages. one might wanna consider replacing the math categories by physics ones. (IMHO, it certainly doesn't belong in the functional analysis category) Mct mht 05:53, 9 September 2006 (UTC)[reply]

i've removed the article from the functional analysis category. Mct mht 20:45, 23 September 2006 (UTC)[reply]

Basis of a vector space

The point of a basis is that it is a minimal (non-redundant, i.e. linear independent) set of vectors or functions that span a certain space.

E.g. for vectors could be defined as follows:

A system of vectors $v_{1},\ldots ,v_{r}\in V$ is called Basis of $V$ if they are a spanning set of $V$ and they are linearly independent.

1. $v_{1},\ldots ,v_{r}\in V$ is a spanning set of a vector space $V$ if $V=span(v_{1},...,v_{r})=\left\{{\lambda _{1},v_{1}+\cdots +\lambda _{r}v_{r}|\lambda _{1},\ldots ,\lambda _{r}\in \mathbb {K} }\right\}$

2. Linear independence: $v_{1},\ldots v_{r}$ are linearly independent if $\lambda _{1}v_{1}+\cdots \lambda _{r}v_{r}=0\Rightarrow \lambda _{1}=\cdots \lambda _{r}=0$ .

These statements are all equivalent: - The vectors $v_{1},\ldots v_{r}$ are a basis.

- Every vector in $V$ is defined by a unique linear combination of the basis vectors.

- If you take out one vector from the basis $v_{1},\ldots v_{r}$ , it is no longer a spanning set of $V$ .

... And of course the number of basis vectors coinsides with the dimension of the vector space.

—The preceding unsigned comment was added by 195.176.0.51 (talk • contribs) 01:25, 16 September 2006 (UTC)