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Unformatted text preview: Chapter 2 Linear Transformations 2.1 Introduction In section 1.3, we used the terms row vector and column vector to describe a matrix which has only one row or column. The sets Mat(1,n) and Mat(n,1) of all 1 × n row vectors, n × 1 column vectors respectively (and more generally the set Mat(n,m) of n × m matrices for any n and m ,) carry a very important algebraic structure, which we now describe: Definition 1 A set V is an Rvector space if the following conditions are satisfied: i) There is an addition operation + on V × V , which to each pair ( v 1 , v 2 ) in V × V assigns an element v 1 + v 2 in V. ia) The operation + is commutative , i.e. v 1 + v 2 = v 2 + v 1 for all v 1 , v 2 ∈ V , and associative , i.e. ( v 1 + v 2 ) + v 3 = v 1 + ( v 2 + v 3 ) for any v 1 , v 2 , v 3 ∈ V . ib) There is a unique element 0 in V , called the zero vector , such that 0 + v = v for any v ∈ V , and each element v ∈ V has a unique additive inverse v , such that v + v = 0 . ii) There is a scalar multiplication defined from R × V to V , where ( r, v ) ∈ R × V 7→ rv ∈ V. iia) Scalar multiplication is distributive over + , i.e. for any r ∈ R , v 1 , v 2 ∈ V we have r ( v 1 + v 2 ) = rv 1 + rv 2 , 1 and for any r 1 , r 2 ∈ R , v ∈ V , we have ( r 1 + r 2 ) v = r 1 v + r 2 v. iib) Scalar multiplication is associative , so that r 1 ( r 2 v ) = ( r 1 r 2 ) v for any r 1 , r 2 ∈ R , v ∈ V . iic) 1 v = v for any v ∈ V . 2 The elements of a vector space V are called vectors . Thus a vector space is a set which has two operations defined on it, addition and scalar multiplication, and these operations satisfy various properties which are compatible with addition and multiplication in the real numbers R . In fact, the reader should immediately verify that R itself is a vector space over R , i.e if we take V = R in Definition 1 then each of the statements in Definition 1 are true. The real numbers are, in some sense, the simplest possible nontrivial vector space over the real numbers, i.e. the simplest vector space over R which is not the trivial vector space { } , which contains just a single vector 0. The sense in which we mean “simplest” is made precise in the following Definition 2 Let V be a vector space. A subset W of V is called an Rsubspace of V if the following conditions hold: i) W contains the zero vector of V . ii) W is closed under the addition and scalar multiplication operations defined for V , i.e. w 1 + w 2 ∈ W for any w 1 , w 2 ∈ W , and rw ∈ W for any r ∈ R , w ∈ W . We use the notation W ≤ V when W is a subspace of V . Every vector space V has at least one subspace, namely { } , and a natural question to ask is whether there are other nontrivial subspaces. When we said that R was the “simplest” nontrivial vector space over R , we meant that there are no nontrivial Rsubspaces of R . This is easy to see, since if W ≤ R is nontrivial, it must contain a nonzero real number, and this allows us, via scalar multiplication, to obtain every...
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 Spring '08
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 Math, Linear Algebra, Algebra, Transformations, Vectors, Vector Space, Sets, TA

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