RE: Re: Are points co-planar in (numDimensions-1)?
- To: mathgroup at smc.vnet.net
- Subject: [mg43287] RE: [mg43240] Re: Are points co-planar in (numDimensions-1)?
- From: "Ingolf Dahl" <ingolf.dahl at telia.com>
- Date: Sat, 23 Aug 2003 08:09:18 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Use NullSpace!
The following vectors are evidently coplanar:
vectors = {{1., 0., 0.}, {0., 1., 0.}, {1., 1., 0.}};
If you take NullSpace of this, you get the orthogonal complement
(a set of all orthogonal vectors) to this set.
NullSpace[vectors]
gives
Out[2]= {{0., 0., 1.}}
which is orthogonal to all the vectors.
If you take
NullSpace[NullSpace[vectors]]
you get back a set of vectors that span your original set.
This method works for any number of dimensions.
Ingolf Dahl
Sweden
>-----Original Message-----
>From: AngleWyrm [mailto:no_spam_anglewyrm at hotmail.com]
To: mathgroup at smc.vnet.net
>Sent: Thursday, August 21, 2003 04:26
>To: mathgroup at smc.vnet.net
>Subject: [mg43287] [mg43240] Re: Are points co-planar in (numDimensions-1)?
>
>
>My last post on this subject lacked depth, so here's more info.
>
>Given some n-dimensional vectors, are they coplanar in n-1? Let
>a1, a2, ..., an be vectors. If they
>are coplanar, then there exists a set of coefficients {k1, k2,
>..., kn}, not all zero, which satisfy
>the equation:
>
>k1 a1 + k2 a2 + ... + kn an = 0.
>
>I have a dataset I wish to test for this property in various dimensions:
>
>numDimensions = 2;
>dataSet = ReadList["data.txt", Number ];
>dataSet = Partition[ dataSet, numDimensions ];
>
>At this point I have a set of 2D vectors, and if I take any two of
>them they HAVE to be coplanar,
>right?
>
>sample = Take[ dataSet, numDimensions];
>Sum[ k\_i sample[[i]], {i, numDimensions} ] (I've used \_ to
>indicate subscript here)
>
>Now my problem is in solving for zero on the last equation. I've
>tried like so:
>Solve[ Sum[ k\_i sample[[i]], {i, numDimensions} ] ==0, {k\_i} ]
>
>but it gives an empty set every time. Thanks for any help.
>-Jonathan
>