1.14 Bases

Digital signal processing Page 1 / 1

Definition 1

A basis of a vector space

V

is a set of vectors

B

such that

$span (B) = V$ .
$B$ is linearly independent.

The second condition ensures that all bases of $V$ will have the same size. In fact, the dimension of a vector space $V$ is defined as the number of elements required in a basis for $V$ . (Could easily be in infinite.)

$R^{N}$ with $B$ the “standard basis” for $R^{N}$
${b_{1}, b_{2}, ..., b_{N}} = \{[\begin{matrix} 1 \\ 0 \\ ⋮ \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 1 \\ ⋮ \\ 0 \end{matrix}], ..., [\begin{matrix} 0 \\ 0 \\ ⋮ \\ 1 \end{matrix}]\}$
Note that this easily extends to $ℓ_{p} (Z)$ .
$R^{N}$ with any set of $N$ linearly independent vectors
$V = {polynomialsofdegreeatmost p}$ $B = {1, t, t^{2}, . . ., t^{p}}$ (Note that the dimension of $V$ is $p + 1$ )
$V = {f (t) : f (t) isperiodicwithperiod T}$ $B = {e^{j k t}}_{k = - \infty}^{\infty}$ (Fourier series, infinite dimensional)

<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Digital signal processing. OpenStax CNX. Dec 16, 2011 Download for free at http://cnx.org/content/col11172/1.4

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Digital signal processing' conversation and receive update notifications?

Ask

©flickr: Jovial	Video Mid Term By Dewey Compton Start Exam
	2 AP Key Terms 02 Chemical Level of Organization By OpenStax Start Key Terms
	11 AP Key Terms 11 The Muscular System By OpenStax Start Key Terms
	47 Biology 47 Conservation Biology Biodiversity MCQ By OpenStax Start Quiz
	14 AP Key Terms 14 Brain Cranial Nerves By OpenStax Start Key Terms
	2 Gastrointestinal Pathophysiology Self-Assessment By Laurence Bailen Start Assessment
	World Capitals Quiz By Yasser Ibrahim Start Quiz
	6 Microeconomics 06 Elasticity By OpenStax Start Flashcards
	4 Neuroanatomy 04 Spinal Cord Brain Stem By Stephen Voron Start Quiz
	10 Lec:10 Sampling Confidence Intervals By Janet Forrester Start Quiz