 # A Pair of Biorthogonal Bases in the Real Plane

Initializing live version Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

In mathematics, a pair of biorthogonal bases (a basis and its dual basis) can provide a representation for vectors in the plane; this is an alternative to what can be done with a single orthonormal basis. While in an orthonormal basis the basis vectors are mutually orthogonal, in a pair of biorthogonal bases the first vector in the basis (solid black in figure) is orthogonal to the second vector in the dual basis (dashed red in figure). Similarly, the second vector in the basis (dashed black in figure) is orthogonal to the first vector in the dual basis (solid red in figure). Furthermore, the corresponding vectors in the two bases have an inner product equal to 1. As each basis can be represented by a matrix whose columns are the basis vectors, the two bases are given when they exist (which is when the matrices are invertible).

[more]

For an orthonormal basis represented by a matrix , the following is true: ,

while for the two matrices and , representing bases in a biorthogonal pair, the counterpart is .

[less]

Contributed by: Jelena Kovacevic (June 2012)
Open content licensed under CC BY-NC-SA

## Snapshots   ## Details

Reference

 M. Vetterli, J. Kovačević, and V. K. Goyal, Foundations of Signal Processing, Cambridge: Cambridge University Press, 2014. www.fourierandwavelets.org.

## Permanent Citation

Jelena Kovacevic

 Feedback (field required) Email (field required) Name Occupation Organization Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback. Send