Projection onto Linearly Dependent Vectors
Now consider another example:
![\begin{eqnarray*}
\sv_0 &\isdef & [1,1], \\
\sv_1 &\isdef & [-1,-1].
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img916.png)
The projections of
onto these vectors are

The sum of the projections is

Something went wrong, but what? It turns out that a set of
vectors can be used to reconstruct an arbitrary vector in
from
its projections only if they are linearly independent. In
general, a set of vectors is linearly independent if none of them can
be expressed as a linear combination of the others in the set. What
this means intuitively is that they must ``point in different
directions'' in
-space. In this example
so that they
lie along the same line in
-space. As a result, they are
linearly dependent: one is a linear combination of the other
(
).
Next Section:
Projection onto Non-Orthogonal Vectors
Previous Section:
Changing Coordinates