How to Find the Basis of a 23D Subspace Using Maple or Matlab?

[uwowizard]

Hi there,

I have 23D subspace, defined by an equation (hyperplane)
c1*x1 + ... + c24*x24 = 0;

I wonder if there is an automated way to find basis of the subspace? I have access to Maple and Matlab.

Thanks.

 

[micromass]

Hi uwowizard!

You can calcate this easily by hand:

For example, if your hyperplane is $$ax+by+cz=0$$, then one can assume one of a,b,c to be nonzero. Let's say c. A basis would then be

$$\{(1,0,-a/c),(0,1,-b/c)\}$$

For your 23D-subspace, assume that c₂₄ is nonzero, then a basis consists of

$$\{ (1,0,0,...,-c_1/c_{24}),(0,1,0,...,-c_2/c_{24}),...,(0,0,0,...,1,-c_{23}/c_{24}) \}$$.

 

[uwowizard]

Thanks a lot!

 

[Geremia]

Hi uwowizard!

You can calcate this easily by hand:

For example, if your hyperplane is $$ax+by+cz=0$$, then one can assume one of a,b,c to be nonzero. Let's say c. A basis would then be

$$\{(1,0,-a/c),(0,1,-b/c)\}$$

For your 23D-subspace, assume that c₂₄ is nonzero, then a basis consists of

$$\{ (1,0,0,...,-c_1/c_{24}),(0,1,0,...,-c_2/c_{24}),...,(0,0,0,...,1,-c_{23}/c_{24}) \}$$.

Isn't this basically the https://secure.wikimedia.org/wikipedia/en/wiki/Gram%E2%80%93Schmidt_process" ?

 

[micromass]

Not really. For applying the Gram-Schmidt process, you must already have a basis. Here you don't have a basis, you must find one. Secondly, the Gram-Schimdt process outputs an orthonormal basis, but my basis here isn't orthonormal in general...

 

[uwowizard]

Micromass, is there a name for the process that you described?

 

[micromass]

Uuh, probably not...