On 5/1/2012 21:59, Colin J. Williams wrote:
On 01/05/2012 2:43 PM, someone wrote:
[snip]
a = [1 2 3]; b = [11 12 13]; c = [21 22 23].
Then notice that c = 2*b - a. So c is linearly dependent on a and b.
Geometrically this means the three vectors are in the same plane,
so the matrix doesn't have an inverse.
Does it not mean that there are three parallel planes?
They're not parallel because our matrix has rank 2, not 1.
Anyway, have a look at this:
http://en.wikipedia.org/wiki/Parallelepiped#Volume
It follows that our matrix A whose rows are a, b and c represents a
parallelepiped. If our vectors are collinear or coplanar, the
parallelepiped is degenerate, i.e. has volume 0. The converse is also true.
Kiuhnm
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