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Date Created: 09/23/15
MEM633 Lectures 3 Chapter 2 Linear Spaces and Linear Operators 21 Linear Spaces Over a Field M field Let F be a set of elements abc for which the sum ab and the product ab of any two elements a and b ofFare de ned Then F is called a eld ifthe following conditions are satisfied 1 Closure ab and ab are in F 2 Commutative laws ab ba ab ba 3 Associative laws abc abc abc abc 4 Distributive law abcabac 5 Zero and Unity F contains an element 0 and an element 1 such that a0a and 1aa 6 Additive inverse For each a in F there is an element b in F such that ab0 7 Multiplicative inverse For each nonzero a in F there is an element b in F such that ab1 E ring Let F be a set of elements abc for which the sum ab and the product ab of any two elements a and b of F are defined Then F is called a ringif the following conditions are satis ed 1 Closure ab and ab are in F 2 Commutative laws abba abba 3 Associative laws abc abc abc abc 4 Distributive law abcabac 5 Zero and Unity F contains an element 0 and an element 1 such that a0a and 1aa 6 Additive inverse For each a in F there is an element b in F such that ab0 Examples a The set of real numbers R is a field b The set of complex numbers lt3 is a field c The set 01 does not form a field if we use the usual definition of addition and multiplication because the element112 is not in the set 01 However if we de ne 000 110 101 000 010 111 then the set 01 with the de ned operations forms a eld d The set of rational functions with real coefficients is a field L6 7L 5d 9 inlay is A VIM HP nrl o amplll q 7 Slak If Planm AI DALEus w h Ill lial 5 A 33 W ml ELM M a linear space X over a eld F A linear space X over a field F is a set of elements called vectors together with two operations vector addition and scalar multiplication The two operations are defined such that the following conditions are satis ed 1 Any vectors Xy in X determine a vector Xy in X called the sum ofX and y 2 Any vector X in X and any scalar c in F determine a scalar product cx in X 3 Xy yx commutative 4 xyz xyz associative 5 There is a zero vector 0 in X such that x0x for allx in X 6 cXy cxcy distributive 7 abx aXbX distributive 8 abx abx associative 9 0 x 0 1 xx 10 For each X in X there is a vector X in X such that XX 0 Examples a R is a vector space over the eld R b is a vector space over the eld lt2 c is a vector space over the eld R d R is not a vector space over Why e Rn is a vector space over R f n is a vector space over lt3 9 Consider the interval ab on the real line Cab the set of all realvalued continuous functions on ab is a vector space over the field R The addition and scalar multiplication are defined in the usual way Def subspace A nonempty subset M of a linear space X is a subspace of X if M is itself a vector space Theorem A nonempty subset M of a linear space X is a linear subspace of X if Xy is in M wheneverX and y are both in M and if also ex is in M whenever X is in M and c is any scalar in F Mcx 4 A la win unarx K 7 e M 9 xayet 12M oeF LFGM Example 2 In the twodimensional real vector space R every straight line passing through the origin 2 is a subspace of R 22 Linear Independence and Bases Def linear independence A nite set of vectors X1 X2 quot39 x in a linear space X over a field F is said to be linearly dependent if and only if there exist scalars c1 0 c in F not all zero such that 2 1 01x 02x2 c x 0 If the finite set X1 X2 quot39 x is not linearly dependent it is called linearly independent Example s2 XE m x2 s1s3 2 E s3 The set of vectors X1 x2 is linearly independent in the field of real numbers However this set of vectors is linearly depend ent in the field of rational functions with real coefficients Why Def dimension Let X be a vector space Suppose there is some positive integer n such that X contains a set of n vectors that are linearly independent while every set of n1 vectors in X is linearly dependent Then X is called nite dimensional and n is the dimension of Example a Rn is ndimensional b Cab is infinite dimensional Why Def linear space spanned by a set of vectors X is a linear space over F Suppose X is any nonempty subset ofX Consider the set M of all nite linear combinations of elements of 2 Le elements ofthe form c1x7 02 X2 c X where n is any positive integer X1 x2 quot39 X are any elements of Z and c1 c are any scalars in F This set M is a linear space generated or spanned by 2 Def basis A finite set E in a space X is called a basis of X if 2 is linearly independent and if the linear subspace generated by 2 is all of X Theorem In an n dimensional vector space any set of n linearly independent vectors quali ed as a basis Def In an n dimensional vector space X if a basis e7 e2 e is chosen then every vector X in X can be uniquely written in the form 1 2 n X a1e 829 ane e eZe a where a a1 a 39 a is called the representation ofX with respect to the basis 97 e2 e Example with respect to different bases Let x X e7e2e a 2 quot 62 1 Suppose 6 A1 A2 A u el e e 1 2 n r1 e p 9 Le e eZe i 2 iip p2p i then Pa where P p1 p2 p Similarly if we choose i e1ezm en qr i12n then we have a 05 where Q q7 q2 q It is easy to see that PQ 1 or o Pquot Change of Basis A vector X in X has different representations