Introduction 3
but ca n ofte n maintai n som e o f th e interestin g an d usefu l character -
istics o f ONBs .
The propert y tha t make s orthonorma l base s desirabl e i n man y
applications i s that w e can find the (unique ) expansio n coefficient s fo r
a vecto r b y takin g inne r product s (do t product s i n R n o r C n ). Thi s
requires fewer computation s tha n matri x inversion, an d i s numerically
more stable . Let' s sa y yo u wan t t o sen d a signa l acros s som e kin d
of communication system , perhap s b y talkin g o n you r wireles s phon e
or sendin g a phot o t o you r mo m ove r th e internet . W e thin k o f tha t
signal a s a vecto r i n a vecto r space . Th e wa y i t get s transmitte d i s
as a sequenc e o f coefficient s whic h represen t th e signa l i n term s o f a
spanning set . I f tha t spannin g se t i s a n ONB , the n computin g thos e
coefficients jus t involve s finding som e do t product s o f vectors , whic h
a compute r ca n accomplis h ver y quickly . A s a result , ther e i s no t a
significant tim e dela y i n sendin g you r voic e o r th e photograph . Thi s
is a good featur e fo r a communication syste m t o have, so orthonorma l
bases ar e use d a lo t i n suc h situations .
Orthogonality i s a ver y restrictiv e property , though . Wha t i f
one o f the coefficient s representin g a vecto r get s los t i n transmission ?
That piec e of information canno t b e reconstructed. I t i s lost. Perhap s
we'd lik e ou r syste m t o hav e som e redundancy , s o tha t i f on e piec e
gets lost , th e informatio n ca n b e piece d togethe r fro m wha t doe s ge t
through. Thi s i s wher e frame s com e in . Generally , a fram e fo r a
finite-dimensional vecto r spac e i s jus t a spannin g se t fo r th e vecto r
space. I n particular , i t nee d no t b e a basis , whic h woul d requir e
linear independence . Wher e frame s ge t interestin g i s that w e can find
certain frame s tha t retai n tha t ver y hand y ON B propert y - tha t w e
can find th e coefficient s fo r expandin g vector s usin g th e do t produc t
instead of matrix inversion. W e can retain the quick computation tim e
found i n ONB s whil e no t restrictin g ourselve s i n number , norms , o r
linear independence .
By using a frame instea d o f an ONB, we do give up the uniquenes s
of th e coefficient s an d th e orthogonalit y o f th e vectors . I n man y
circumstances, however , thes e propertie s ar e superfluous . I f yo u ar e
sending you r sid e of a phone conversatio n o r a photo, wha t matter s i s
quickly computing a working set of expansion coefficients, no t whethe r
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