I am not understanding how this has dimension $3$, but there are six components in each vector. If $3$ vectors span the space, why are there more than $3$ components in each vector? I thought for a set to be a basis there should be $n$ vectors and $n$ components in the vectors so that we may express the set as a square matrix.
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The space does have 6 dimensions as you pointed out. The points that are solutions in that space, however, only make up a 3 dimensional subspace.
Using some smaller space examples we could look at the line y=x in 2-dimensions and note that the solution space is only 1 dimension (the solution is a 1 dimensional shape (a line) in a 2 dimensional world).
Or if we look at a plane z=x+y it'll be a 2-dimensional shape/solution in a 3-dimensional world.
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$\begingroup$ So, its basically like a 3 dimensional object expressed in a 6 dimensional world? Like a sphere is a 2 manifold but it is expressed in a 3 dimensional world. $\endgroup$– user5826Apr 14, 2014 at 18:19