Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 642:527 FALL 2009 EXPANSIONS IN ORTHOGONAL BASES Vector spaces We will use without much further comment the idea of a vector space . Basically, a vector space is a set of vectors (these vectors will in fact often be functions) with the property that a linear combination of vectors is again a vector. Linear combinations involve scalars , which may be real numbers (for a real vector space ) or complex numbers (for a complex vector space ). If u and v are vectors in a vector space, and α and β are scalars, then αf + βg is also a vector in the space. These linear combinations are required to satisfy a set of rules which any reasonable person would consider obvious. See Section 9.6 of Greenberg, and in particular Definition 9.6.1, for a careful discussion of vector spaces. For the moment we will consider only real vector spaces, returning to complex vector spaces at the end of these notes. Example 1: One familiar vector space is R n , the set of all row vectors with n (real) components. If u = ( u 1 , . . ., u n ) and v = ( v 1 , . . ., v n ) are two vectors in R n then their linear combinations are constructed by making linear combinations of their components: α u + β v = ( αu 1 + βv 1 , . . ., αu n + βv n ) . (1) Example 2: Another example of a vector space, important for the theory of Fourier series and similar applications, is C p [ a, b ], the set of all piecewise continuous, real-valued functions f ( x ) defined for a ≤ x ≤ b . (Piecewise continuity is defined on page 249 of Greenberg.) As in any space of functions, the rule for linear combinations is ( αf + βg )( x ) = αf ( x ) + βg ( x ) , a ≤ x ≤ b. (2) An element f of C p [ a, b ] is of course a function but, since it belongs to a vector space, we may speak of it as a “vector” when we want to emphasize this context. Read Section 17.6 of Greenberg for more on C p [ a, b ] as a vector space. Notation: When we speak of a general vector space in these notes we will denote typical vectors as in Example 1, using boldface letters: u , v , etc., and later e 1 , e 2 , . . . . The reader should bear in mind, however, that what we say applies equally well when the vectors under consideration are functions, considered as members of a vector space like C p [ a, b ]. When we are speaking specifically about functions we will denote them, as in Example 2, by the letters f , g , etc. 1 640:527 EXPANSIONS IN ORTHOGONAL BASES FALL 2009 Inner products An inner product in a vector space is a formula which assigns to any pair of vectors, say u and v , a number ( u , v ) , their inner product. When the vector space is R n there is a familiar inner product (usually called the dot product): ( u , v ) = u · v = n summationdisplay i =1 u i v i . (3) When the vector space is C p [ a, b ] the most common inner product, used in the study of Fourier series, is ( f, g ) = integraldisplay b a f ( x ) g ( x ) dx. (4) Inner products are discussed further in Section 9.6.2 of Greenberg....
View Full Document

This note was uploaded on 12/15/2009 for the course MATH 527 taught by Professor Staff during the Fall '08 term at Rutgers.

Page1 / 9


This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online