Re: Strange result in MMa 3.0

• To: mathgroup at smc.vnet.net
• Subject: [mg8201] Re: Strange result in MMa 3.0
• From: Daniel Lichtblau <danl>
• Date: Mon, 18 Aug 1997 23:24:50 -0400
• Organization: Wolfram Research, Inc.
• Sender: owner-wri-mathgroup at wolfram.com

```Julian Stoev !!! Address is anti-spamed !!! wrote:
>
> Dear Group,
> I am used to think, that the definition of NullSpace is:
> NullSpace of linear operator A is a set N(A) defined by all elements x for
> which A.x=[0]
>
> But Mathematica 3.0 gives me this strange result:
> =========
> In[1]:= A = {{0, 1, 1, 2, -1}, {1, 2, 3, 4, -1},{2, 0, 2, 0, 2}}
> Out[1]= {{0, 1, 1, 2, -1}, {1, 2, 3, 4, -1},{2, 0, 2, 0, 2}}
>
> In[2]:= NullSpace[A]
> Out[2]= {{-1, 1, 0, 0, 1}, {0, -2, 0, 1, 0},{-1, -1, 1, 0, 0}}
> (*this is correct, but transposed and this makes problems later*)
>
> In[3]:=
> A . Transpose[NullSpace[A]]
> Out[3]=
> {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}
> ===========================
> The problem is that in Mathematica A . transpose(nullspace(A))=[0].
> I don't want to give here the result when I try correct A . Nullspace(A)
> The program reports that tensors have incopatible dimensions and returns
> unevaluated result.
> So I was wondering why Mathematica gives result for NullSpace transposed?
> Is the definition I am using for NullSpace wrong?
>
> Thanks for any hints where is the problem!
> _______________________________________________________________________
> Julian  Stoev - Ph.D. student          |phone: Work 872-7283
> Intelligent Information Processing Lab.|       Home 741-4630 (r.442)
> Seoul National University              |       http://poboxes.com/stoev
> -----------------------------------------------------------------------
> !!!!! to contact me, remove anti-spam in FROM:

As you note, NullSpace[A] returns a list of vectors that span the null
space of A. Each null vector is an element of this list. To obtain a
matrix B such that A.B is all zeroes, one would have to transpose
because now you'd want the null vectors to be columns of your matrix. In
other words, A.NullSpace[A] should NOT give zeroes, because the null
space is a list of null (row) vectors, not a matrix of null column
vectors.

Daniel Lichtblau
Wolfram Research

```

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