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== Extension to polynomials? ==
When working with polynomials and an arbitrary weight function, there is a recursive Gram-Schmidt orthonomalization technique. More details are provided below:

http://mathworld.wolfram.com/Gram-SchmidtOrthonormalization.html
[[Special:Contributions/70.162.89.24|70.162.89.24]] ([[User talk:70.162.89.24|talk]]) 05:51, 30 August 2013 (UTC)

:Another example or two is needed, particularly something in a function (ie functional analysis) setting. For example, some simple polynomial examples with a suitable inner product. For example,

::<math>\mathbf{u}_1 = 1 </math>,   <math>\mathbf{u}_2 = t</math>, <math>\mathbf{u}_2 = t^2, \dots</math>. 

:One covers examples like this in introductions to signal processing, so I imagine it's quite important in certain engineering fields. Also, this will produce orthogonal or orthonormal polynomials, making a nice connection to special functions. For example on the interval <math>[-1,1]</math> with the inner product <math>\int f(t) g(t) dt</math>, one recovers the [[Legendre_polynomials#Orthogonality|Legendre polynomials]].[[User:Improbable keeler|Improbable keeler]] ([[User talk:Improbable keeler|talk]]) 09:29, 18 January 2018 (UTC)

== Determinant formula ==
I have two complaints about this section!

First of all, the section defines some vectors ''e<sub>i</sub>'' and then never mentions them again. Frankly, I have no idea what's going on there.

Secondly, the section contains the text: "Note that the expression for uk is a "formal" determinant, i.e. the matrix contains both scalars and vectors; the meaning of this expression is defined to be the result of a cofactor expansion along the row of vectors."

I highly doubt that this explanation will be the least bit enlightening to anyone who didn't already know what was going on, and I also doubt that the link would be very enlightening with a bit of additional context. (For instance, the text should indicate which row the Laplace expansion is being taken over - the last one - and then indicate the formula as a linear combination of the ''v'''s  with coefficients combing from the corresponding minor determinants.) [[Special:Contributions/2602:30A:C04C:5F30:805:108B:A61A:2175|2602:30A:C04C:5F30:805:108B:A61A:2175]] ([[User talk:2602:30A:C04C:5F30:805:108B:A61A:2175|talk]]) 23:37, 3 August 2014 (UTC)

Francesco Caravelli: I have been trying very hard to find the determinant formula in the literature. There is no reference in the wiki page. Very frustrating! 
Update: I have found a similar formula in Gantmacher: Theory of matrices (1959) Volume 1, Pages 256-258.

 <!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/204.121.137.208|204.121.137.208]] ([[User talk:204.121.137.208#top|talk]]) 17:28, 16 March 2017 (UTC)</small> <!--Autosigned by SineBot-->

==Untitled==
The Matlab implementation for the Gram-Schmidt process is for a specific norm and inner product definition (here being the Standard Euclidean Inner Product and by it's extension the 2-norm). 
Should be updated to reflect that.   <!-- Template:Unsigned --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:BlackMetalStats|BlackMetalStats]] ([[User talk:BlackMetalStats#top|talk]] • [[Special:Contributions/BlackMetalStats|contribs]]) 00:19, 3 April 2017 (UTC)</small> <!--Autosigned by SineBot-->

==Historical origins==
The article originally said that the method had appeared in the work by both Laplace and [[Augustin-Louis Cauchy|Cauchy]], citing the Cheney and Kincaid book. <ref>{{cite book |last=Cheney |first=Ward |last2=Kincaid |first2=David |title=Linear Algebra: Theory and Applications |location=Sudbury, Ma |publisher=Jones and Bartlett |year=2009 |url={{Google books |plainurl=yes |id=Gg3Uj1GkHK8C |page=544 }} |isbn=978-0-7637-5020-6 |pages=544, 558 }}</ref> But the book only mentions that Laplace was "familiar" with the method. I couldn't see a reference to Cauchy on Google Books, but I don't have a copy of the book. Arguably a better historical reference is needed.[[User:Improbable keeler|Improbable keeler]] ([[User talk:Improbable keeler|talk]]) 06:57, 18 January 2018 (UTC)

{{reflist-talk}}

== The definition of projection is INCORRECT+Proof ==
If we consider <math>proj_{n}(v) := \frac {<n,v>}{<n,n>} n</math>, I proof the set you get isn't orthogonal to ech other. Let's proof:

For the field we consider Complex Numbers. If <math>proj_{n}(v) := \frac {<n,v>}{<n,n>} n </math> then we begin by compute <math> u_i </math>.

<math display="block"> u_1 = v_1 </math>
<math display="block"> u_2 = v_2 - proj_{u_1}(v_2)</math>

Now see what happen when <math> <u_2,u_1> </math>:

<math display="block"><u_2,u_1>=<v_2-proj_{u_1}(v_2),u_1>=<v_2-proj_{v_1}(v_2),v_1> = <v_2,v_1>-<proj_{v_1}(v_2),v_1>
= <v_2,v_1> - < \frac {<v_1,v_2>}{<v_1,v_1>} v_1 , v_1 > </math>

Now the <math> \frac {<n,v>}{<n,n>} </math> is number in field so we can get it out of bracket as define <ref>{{Citation|title=Inner product space|date=2019-04-30|url=https://en.wikipedia.org/w/index.php?title=Inner_product_space&oldid=894916386|work=Wikipedia|language=en|access-date=2019-05-15}}</ref>.

<math> <v_2,v_1>-\frac {<v_1,v_2>}{<v_1,v_1>} < v_1 , v_1 > = <v_2,v_1>-<v_1,v_2> </math>
and that ISN'T Zero in Complex Vector Space; But if you write the correct form which is <math> proj_{n}(v) := \frac {<v,n>}{<n,n>} n </math> every think make sense.

[[User:Firouzyan|Firouzyan]] ([[User talk:Firouzyan|talk]]) 21:35, 15 May 2019 (UTC)

I believe in the above there's a missing complex conjugation.

<math>< \frac {<v_1,v_2>}{<v_1,v_1>} v_1 , v_1 > = \left(\frac {<v_1,v_2>}{<v_1,v_1>}\right)^* <v_1,v_1></math>
I've edited accordingly (changing it back to the pre-2019 state, I think).

[[User:Bytbox|Scott Lawrence]] ([[User talk:Bytbox|talk]]) 03:37, 5 September 2024 (UTC)

Apologies; I've just realized that I've run into this difference of convention between physics and the rest of the world. Changing the page back, but I'll add a short note somewhere to save other poor souls the effort.

[[User:Bytbox|Scott Lawrence]] ([[User talk:Bytbox|talk]]) 03:52, 5 September 2024 (UTC)

{{reflist-talk}}