Multiple Coordinate Spaces - 3D Math Primer for Graphics and Game Development - page 102

Figure 3.12
The two basis vectors p and q
span a 2D subset of the 3D
space.
Consider the vector c , which lies behind the plane in Figure 3.12. This
vector is not in the span of p and q , which means we cannot express it as
a linear combination of the basis. In other words, there are no coordinates
[c x ,c y ] such that c = c x p + c y q .
The term used to describe the number of dimensions in the space spanned
by the basis is the rank of the basis. In both of the examples so far, we
have two basis vectors that span a two-dimensional space. Clearly, if we
have n basis vectors, the best we can hope for is full rank, meaning the
span is an n-dimensional space. But is it possible for the rank to be less
than n? For example, if we have three basis vectors, is it possible that the
span of those basis vectors is only 2D or 1D? The answer is “yes,” and this
situation corresponds to what we meant earlier by a “poor choice” of basis
vectors.
For example, let's say we add a third basis vector r to our set p and q . If
r lies in the span of p and q (for example, let's say we chose r = a or r = b
as our third basis vector), then the basis vectors are linearly dependent,
and do not have full rank. Adding in this last vector did not allow us to
describe any vectors that could not already be described with just p and
q . Furthermore, now the coordinates [x,y,z] for a given vector in the span
of the basis are not uniquely determined. The basis vectors span a space
with only two degrees of freedom, but we have three coordinates. The
blame doesn't fall on r in particular, he just happened to be the new guy.
We could have chosen any pair of vectors from p , q , a , and b , as a valid
basis for this same space. The problem of linear dependence is a problem
with the set as a whole, not just one particular vector. In contrast, if our