I think echelonize is a bit misleading, because it's not really doing 
Gaussian elimination. Besides, I think it would have to be rref to be 
consistent with matrices (it's reduced row echelon form, not simply row 
echelon form):

sage: matrix([0,3,4]).echelon_form()
[0 3 4]
sage: matrix([0,3,4]).rref()
[  0   1 4/3]

Even if "normal form" is used in some specific contexts, I don't think many 
people would associate it with that function, and it might create confusion 
with normalize().

How about a vote?

   1. reduce() (kind of ambiguous and might be mistaken for the Python 
   function reduce)
   2. echelonize() (probably echelon_form() is more appropriate, since it's 
   not mutating the vector)
   3. rref()
   4. normal_form()
   5. monicize()
   6. divide_by_leading()

I vote (6).

On Friday, 30 November 2012 14:15:34 UTC-8, kannappan sampath wrote:
>
> Here's a "natural" place where I can see this is helpful: 
>
> To carry out, Gauss-Jordan like row reductions, where you'd like the first 
> non-zero entry in a row (the so-called pivot) to be 1, I think this is very 
> helpful.
>
> So, I'd be **for** keeping this, perhaps in a different name, that would 
> reflect this. Like `reduce(v)` or perhaps, `echelonize(v)` for instance. 
>
> Regards, 
> ~KnS
>
>
> On Sat, Dec 1, 2012 at 3:06 AM, Eviatar <eviat...@gmail.com 
> <javascript:>>wrote:
>
>> Hello,
>>
>> I was planning to work on the patch for this; however, is "normal form" 
>> known terminology for dividing a vector by the leading entry? A Google 
>> search returns no such use. Wouldn't it be better for it to be named 
>> something like divided_by_leading, to remove ambiguity and potential 
>> confusion?
>>
>> On Wednesday, 19 September 2012 11:55:24 UTC-7, Chris wrote:
>>>
>>> It was pointed out to me that vector normalization in Sage is not doing 
>>> what most people would think it should do. Rather than divide by the norm 
>>> of the vector, right now the implementation is to divide the vector by the 
>>> first non-zero part. This was pointed out to me by someone who teaches 
>>> undergrads and found it frustrating that Sage was not doing the standard 
>>> normalization. I think it needs to be changed.
>>>
>>> Anyways, I have opened a trac ticket, take a look:
>>>
>>> http://trac.sagemath.org/sage_**trac/ticket/13393<http://trac.sagemath.org/sage_trac/ticket/13393>.
>>>  
>>>
>>>
>>> It was suggested that this come to a vote. I vote that it be changed. 
>>> Opinions anyone?
>>>
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