Projection onto Linearly Dependent Vectors

Chapter Contents:

Search Mathematics of the DFT

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

Projection onto Linearly Dependent Vectors

Now consider another example:

$\begin{eqnarray*} \sv_0 &\isdef & [1,1], \\ \sv_1 &\isdef & [-1,-1]. \end{eqnarray*}$

The projections of onto these vectors are

$\begin{eqnarray*} {\bf P}_{\sv_0}(x) &=& \frac{x_0 + x_1}{2}\sv_0, \\ {\bf P}_{\sv_1}(x) &=& -\frac{x_0 + x_1}{2}\sv_1. \end{eqnarray*}$

The sum of the projections is

$\begin{eqnarray*} {\bf P}_{\sv_0}(x) + {\bf P}_{\sv_1}(x) &=& \frac{x_0 + x_1}... ... + x_1}{2} (-1,-1) \\ &=& \left(x_0+x_1,x_0+x_1\right) \neq x. \end{eqnarray*}$

Something went wrong, but what? It turns out that a set of vectors can be used to reconstruct an arbitrary vector in ${\bf C}^N$ 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 ( ).

Order a Hardcopy of Mathematics of the DFT

Previous: An Example of Changing Coordinates in 2D
Next: Projection onto Non-Orthogonal Vectors

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

Sign in

Search Online Books

Free Online Books

Sponsor

Chapters

Mathematics of the DFT
    Geometric Signal Theory
       Signal Reconstruction from Projections
          Projection onto Linearly Dependent Vectors