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MathGroup Archive 2003

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Re: Rows & Columns; What do *You* Call a Vector

  • To: mathgroup at smc.vnet.net
  • Subject: [mg43950] Re: [mg43924] Rows & Columns; What do *You* Call a Vector
  • From: Sseziwa Mukasa <mukasa at jeol.com>
  • Date: Wed, 15 Oct 2003 04:59:26 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Vectors are rank 1 entities, the quantity c you describe below is a 
rank 2 object, they are not the same.  One dimensional lists in 
Mathematica behave as rank 1 tensors, also known as vectors.

The 2D input methods are for entering rank 2 objects.  They are 
perfectly fine for doing math, as long as you know that you want to 
work with rank 2 objects.

Coincidentally, if T is a m by n rank 2 object of mixed type (a matrix) 
operating on C a n by 1 rank 2 object of mixed type (what you call a 
column vector) to generate R a m by 1 tensor (T.C=R) and v is an n 
dimensional rank 1 tensor where v(m) = C(m,1) and R2 = T.v then R2(m) = 
R(m,1).  Note that R2 is not the same type of object as R.  The 
difference becomes apparent when you try to form C.T, which is only 
defined when m = 1 for the dimensions to match.  v.T on the other hand 
is defined whenever m = n.  This coincidence I think leads to your 
confusion.

Regards,

Ssezi

On Sunday, October 12, 2003, at 04:03 AM, Steven T. Hatton wrote:

> When I create a column vector explicitly using the Mathematica 2D input
> method, the result is (assume '(' and ')' are hight-adjusted),
>
>    c1
> c=(c2)-> {{c1},{c2},{c3}}.
>    c3
>
> If I create a list of length 3 and apply MatrixForm, the result is
>
> v={v1,v2,v3};
>
>                 v1
> MatrixForm[v]->(v2).
>                 v3
>
> It appears that in matrix multiplication involving traditional 
> (euclidian)
> matrix operations, /v/ above behaves as a a row vector when prefixed, 
> and a
> column vector when postfixed.  /c/ above is clearly distinct in form 
> from
> v/ above.  How do others approach this situation?  Do you simply work 
> with
> 'raw' lists, and avoid using the 2D input methods?  Is there a 
> convention
> for using the 'vectors' produced by the 2D input methods, such as
> List[List[r1,r2,r3]]<=>row vector, and
> List[List[c1],List[c2],List[c3]]<=>column vector?
>
> The reason I ask is because when I first began using Mathematica, I
> attempted to use the 2D input vectors, and found the results to be
> different than expected.  I've also noticed that /mmf = MatrixForm[m]/
> tends to behave differently from /m/ in some cases.  Unless there is 
> some
> underlying unifying principle, and/or, convention that makes these 
> various
> forms interoperate, it seems the 2D input methods for creating vectors
> should be avoided except where they are for purly textual purposes, 
> i.e.,
> they should not be used for doing math.
>
> -- 
> Gelonus, Kucha, Kizil, Sampul, Dunhuang, Kanishka, Cherchan, Margiana,
> Sintashta, Himmelsscheibe von Nebra, Ryazan Russia,  Afanasevo, Peñon
> Woman, Guancha, Maes Howe, Purushkhanda.
>


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