1.3.3 Basis sets and Gram-Schmidt Orthogonalization
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1 .. Basis sets and Gram-Schmidt Orthogonalization Before we address the question of existence and uniqueness, we must estalish one more tool for working with ectors asis sets. Let R, with (..-) : We can oiously define the set of unit ectors e [] : e [] : e [] (..-) : so that we can write as e [] + e [] + + e [] (..-) As any R can e written in this manner, the set of ectors {e [], e [], e [] } are said to form a asis for the ector space R. The same function can e performed y any set of mutually orthogonal ectors, i.e. a set of ectors { [], [],, [] } such that [j] if j k (..-4) This means that each [j] is mutually orthogonal to all of the other ectors. We can then write any R as [] [] [] e e +... e + + (..-5) [] Where we use a prime to denote that 9 deg. (..-6) when comparing the expansions (..-) and (..-5) j j [] 9 deg.
2 Orthogonal asis sets are ery easy to use since the coefficients of a ector R in the expansion are easily determined. We take the dot product of (..-5) with any asis ector, k [,], [] ( ) ( ) ( ) (..-6) k [] Because with [j] jk ( )δ δ (..-7) jk, j k δ jk (..-8), j k then (..-6) ecomes k k (..-9) In the special case that all asis ectors are normalized, i.e. for all k [,], we hae an orthonormal asis set, and the coefficients of R are simply the dot products with each asis set ector. Exmaple..- Consider the orthogonal asis for R [] [] (..-) for any R, what are the coefficients of the expansion [] [] + + (..-)
3 First, we check the asis set for orthogonality [] [] [ ] ()() + ()(-) + ()() [] [ ] ()() + ()() + ()() (..-) [] [ - ] ()() + (-)() + ()() We also hae [] [ ] [] [ - ] [ ] (..-) So the coefficients of are [] [] [ ] ( + ) [] [] [ ] ( - ) [ ]
4 Although orthogonal asis sets are ery conenient to use, a set of ectors B { [], [],, [] } need not e mutually orthogonal to e used as a asis they need merely e linearly independent. Let us consider a set of M ectors [], [],, [M] R. This set of M ectors is said to e linearly independent if c [] + c [] + + c M [M] implies c c c M (..-6) This means that no [j], j [,M] can e written as a linear comination of the other M- asis ectors. For example, the set of ectors for R [] [] (..-7) is not linearly independent ecause we can write as a linear comination of [] and [], [] - [] - (..-8) Here, a ector R is said to e a linear comination of the ectors [],, [M] R if it can e written as [] [] [M] +... M + + (..-9)
5 We see that the ectors of (..-7) do not form a asis for R since we cannot express any ector R with as a linear comination of { [], [], } since + + (..-) + We see howeer that if we instead had the set of linearly independent ectors [] [] (..-) then we could write any R as + + (..-) + (..-) defines a set of simultaneous linear equations + (..-) that we must sole for,,, ) (,, (..-4)
6 We therefore make the following statement: Any set B of linearly independent ectors [], [],, [] R can e used as a asis for R. We can pick any M suset of the linearly independent asis B, and define the span of this suset { [], [],, [M] } B as the space of all possile ectors R that can e written as c [] + c [] + + c M [M] (..-5) For the asis set (..-), we choose [] and.(..-6) Then, span { [], } is the set of all ectors R that can e written as c c [] + c c + c (..-7) c Therefore, for this case it is easy to see that span { [], }, if and only if ( iff ). ote that if span{ [], } and w span{ [], }, then automatically + w span { [], }. We see then that span{ [], } itself satisfies all the properties of a ector space identified in section... Since span{ [], } suspace of R. R (i.e. it is a suset of R ), we call span{ [], } a
7 This concept of asis sets also lets us formally identify the meaning of dimension this will e useful in the estalishment of criteria for existence/uniqueness of solutions. Let us consider a ector space V that satisfies all the properties of a ector space identified in section... We say that the dimension of V is if eery set of + ectors [], [],, [+] V is linearly independent and if there exists some set of linearly independent ectors [],, [] V that forms a asis for V. We say then that dim(v). (..-8) While linearly independent asis sets are completely alid, they are more difficult to use than orthogonal asis sets ecause one must sole a set of linear algeraic equations to find the coefficients of the expansion [] [] [] (..-9) : [] [] [] [] [] : [] : [] [] [] : : O( ) effort to sole for all j s (..-) This requires more effort for an orthogonal asis { [],, [] } as j [j] [j] [j] O( ) effort to find all j s (..-9, repeated)
8 This proides an impetus to perform Gramm-Schmidt orthogonalization. We start with a linearly independent asis set { [], [],, [] } for R. From this set, we construct an orthogonal asis set { [], [],, [] } through the following procedure:. First, set [] [] (..-) []. ext, we construct [] such that. Since [] [], and [] and [] are linearly independent, we can form an orthogonal ector [] from [] y the following procedure: [] [] c [] [] sutract this part from [] [] [] Then, taking the dot product with [], We write [] [] + c [] (..-) [] [] [] [] [] [] + c (..-) Therefore c [] [] [] (..-4) And our nd ector in the orthogonal asis is [] [] [] - [] [] [] (..-5)
9 . We now form in a similar manner. Since [] is a linear comination of [] and [], we can add a component from direction to form, + c [] + c [] (..-6) [] First, we want [] [] [] + c + c (..-7) [] [] so c [] [] (..-8) A similar condition that yields [] c [] [] (..-9) so that the rd memer of the orthogonal asis set is [] [] - [] [] [] [] (..-4) 4. Continue for [j], j 4, 5,, where [j] [j] j- [j] - k k (..-4) 5. ormalize ectors if desired (we can do this also during construction of orthogonal asis set) [j] [j] (..-4) [j]
10 As an example, let us use this method to generate an orthogonal asis for R such that the st memer of the asis set is [] (..-4) First, we write a linearly independent asis that is not, in general, orthogonal. For example, we could choose [] [] (..-44) We now perform Gram-Schmidt orthogonalization,. [] [] (..-45). We next set [] [] - [] [] [] [] (..-5, repeated) [] [ ] (..-46) [] [] [ ] (..-47) so
11 [] - (..-48) ote [] [] [/ -/ ] ½ - ½ (..-49) We now calculate - [] [] [] [] [] [] (..-4, repeated) [] [/ -/ ] (..-5) [] [ ] (..-5) [] [ ] (..-5) We therefore hae merely (..-5)
12 [] [] (..-54) Our orthogonal asis set is therefore
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