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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