Re: Are points co-planar in (numDimensions-1)?
- To: mathgroup at smc.vnet.net
- Subject: [mg43283] Re: Are points co-planar in (numDimensions-1)?
- From: Olaf Rogalsky <Olaf.Rogalsky at physik.uni-erlangen.de>
- Date: Sat, 23 Aug 2003 08:09:12 -0400 (EDT)
- Organization: University of Erlangen, Germany
- References: <bi195e$akp$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
AngleWyrm wrote:
> 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 don't agree. This is the definition of linear independence of n vectors, not of coplanarity.
A set of k vectors {a_i in Vn | i=1..k} in a n-dimensional vector space Vn are said to be coplanar,
if all k vectors {a_i in Vn | i=1..k} are elements of a (n-1)-dimensional affine subspace U of Vn.
It is easy to show, that if n==k, any set of vectors {a_i in Vn | i=1..n} is coplanar.
> 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} ]
You have to specify a set of variables to solve for: ^^^^^^
Solve[ Sum[ k\_i sample[[i]], {i, numDimensions} ] ==0, Table[k\_i,{i,1,numDimensions}] ]
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