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Unformatted text preview: Results in Linear Mathematics (P1) T.W.K orner May 19, 2003 Small print The syllabus for the course is defined by the Faculty Board Schedules (which are minimal for lecturing and maximal for examining). I have starred certain results which seem to me to go beyond a strict interpretation of the syllabus. However whilst it would not, in my opinion, be fair to set such results as bookwork they could well appear as problems. I should very much appreciate being told of any corrections or possible improvements. This document is written in L A T E Xand stored in the file labeled ~twk/1B/V1.tex on emu in (I hope) read permitted form. My e-mail address is twk . 1 Vector Spaces Convention 1.1 We shall write F to mean R or C . Definition 1.2 We call ( V, + ,., F ) a vector space over F if, whenever u,v,w V and , F , then u + v V , u V and (i) ( V, +) is an Abelian group (so in particular u +( v + w ) = ( u + v )+ w , u + v = v + u ). (ii) ( u ) = ( ) u . (iii) ( + ) u = u + u . (iv) ( u + v ) = u + v . (v) 1 u = u . Lemma 1.3 (i) The zero of ( V, +) satisfies u = 0 for all u V . (ii) The additive inverse- u of u V satisfies- u = (- 1) u . We call 0 the zero vector and write it as 0. (Our general policy of dropping boldface u and underline u in favour of the simple u will not usually lead to ambiguity but if it does we simply revert to the less simple convention.) 1 Theorem 1.4 If X is any set then the set F X of functions f : X F is a vector space if we define vector addition and multiplication by a scalar by ( f + g )( x ) = f ( x ) + g ( x ) , and ( f )( x ) = f ( x ) for all x X where f,g F X and F . Definition 1.5 If V is a vector space we say that U V is a subspace of V if U and ( , F , u,v U ) u + v U. Lemma 1.6 If U is a subspace of a vector space V then U is itself a vector space. It is usually easier to use Theorem 1.4 (or its generalisation Theorem 2.7 below) together with Lemma 1.6 to prove that something is a vector space than to verify the axioms in Definition 1.2. Example 1.7 The space C ([0 , 1]) of continuous functions f : [0 , 1] F , the space P of real polynomials P : R R , the classical spaces F n and the set J of n n real matrices all of whose rows and columns add up to the same number can all be made into vector spaces in a natural way. Definition 1.8 (i) Vectors e 1 ,e 2 ,... ,e n span a vector space E if given any e E we can find 1 , 2 ,... , n F such that e = 1 e 1 + 2 e 2 + ... + n e n . (ii) Vectors e 1 ,e 2 ,... ,e n in a vector space E are linearly independent if the only solution of 0 = 1 e 1 + 2 e 2 + ... + n e n (with 1 , 2 ,... , n F ) is 1 = 2 = ... = n = 0 ....
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