571_3.4 - 3.4 BASIS AND DIMENSION KIAM HEONG KWA p 138 A...

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Unformatted text preview: 3.4 BASIS AND DIMENSION KIAM HEONG KWA p. 138 A finite set of elements B = { v 1 , v 2 , ยทยทยท , v n } of a linear space V is said to form a basis of V if (1) S is a linearly independent set and (2) V = Span( S ) . pp. 139-140 Let V be a linear space. If there is a spanning set S = { v 1 , v 2 , ยทยทยท , v n } of V having n elements, then any subset of V having m elements, where m > n , must be linearly dependent. Theorem 3.4.1 Proof. Let T = { u 1 , u 2 , ยทยทยท , u m } โŠ† V . Since V = Span( S ) , there is an element a i = ( a i 1 ,a i 2 , ยทยทยท ,a in ) T in R n such that u i = n X j =1 a ij v j = a i 1 v 1 + a i 2 v 2 + ยทยทยท + a in v n for each i โˆˆ { 1 , 2 , ยทยทยท ,m } . It follows that T is linearly dependent if (and only if) there is a c = ( c 1 ,c 2 , ยทยทยท ,c m ) T โˆˆ R m , c 6 = , such that m X i =1 c i u i = m X i =1 c i n X j =1 a ij v j = n X j =1 m X i =1 a ij c i v j = . We claim that such a vector c exists. To see this, consider the n ร— m homogeneous system of linear equations m X i =1 a ij c i = a 1 j c 1 + a 2 j c 2 + ยทยทยท + a mj c m = 0 , j = 1 , 2 , ยทยทยท ,n. In matrix notation, it reads A c = , where A = a 11 a 21 ยทยทยท a m 1 a 12 a 22 ยทยทยท a m 2 . . . . . . . . . a 1 n a 2 n ยทยทยท a mn . Date : July 1, 2011. 1 2 KIAM HEONG KWA WATCH OUT for the way the entries of A are labeled in this case. The system A c = is an underdetermined homogeneous system and has therefore infinitely many solutions as we have shown in section 1.2. In particular, it has a solution c 6 = , which is what to be shown. As a consequence of this theorem, it follows that if a linear space V has a finite basis, then all bases of V has the same number of elements. Corollary 3.4.2 Proof. Let B 1 and B 2 be two bases of V having m and n elements. Since V = Span( B 1 ) , n โ‰ค m ; otherwise, B 2 would be linearly dependent. (Recall that a basis must be linearly independent.) Like- wise, since V = Span( B 2 ) , m โ‰ค n . Hence m = n . In view of this corollary, it makes sense to define the dimension Dimension of a linear space V having a finite basis as the number of elements in the basis, denoted dim( V ) . It is a convention to define the dimen- sion of the linear space { } as . Also, if a linear space has finite dimension, then we say that it is finite dimensional ; otherwise, we say that it is infinite dimensional . Example 1. The polynomial space P consisting of all polynomials of finite degrees is an infinite dimensional linear space under stan- dard addition and scalar multiplication. Suppose not, so that it has a finite dimension n . On the other hand, recall that { 1 ,x,x 2 , ยทยทยท ,x n } is a linearly independent subset of P having n + 1 elements. This contradicts theorem 3.4.1. A similar argument indicates that the function space C [ a,b ] of all continuous functions on a closed interval is also an infinite dimensional linear space....
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571_3.4 - 3.4 BASIS AND DIMENSION KIAM HEONG KWA p 138 A...

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