Re: Strange result in MMa 3.0

• To: mathgroup at smc.vnet.net
• Subject: [mg8197] Re: Strange result in MMa 3.0
• From: Robert Knapp <rknapp>
• Date: Mon, 18 Aug 1997 23:24:46 -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?

No actually both are right.  Think of the list
Out[2]= {{-1, 1, 0, 0, 1}, {0, -2, 0, 1, 0},{-1, -1, 1, 0, 0}}
as representing a set of three elements (vectors),
{-1, 1, 0, 0, 1}, {0, -2, 0, 1, 0}, and {-1, -1, 1, 0, 0},
each of which is in the Null space of A:

In[2]:= A.{-1, 1, 0, 0, 1}
Out[2]= {0, 0, 0}
In[3]:= A.{0, -2, 0, 1, 0}
Out[3]= {0, 0, 0}
In[4]:= A.{-1, -1, 1, 0, 0}
Out[4]= {0, 0, 0}

Or alternatively, you can write

In[5]:= (A.#)& /@ NullSpace[A]
Out[5]= {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}

which is equivalent to A . Transpose[NullSpace[A]].

```

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