This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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...
View
Full
Document
This note was uploaded on 04/17/2008 for the course MATH 21 taught by Professor Marks during the Spring '08 term at UCSC.
 Spring '08
 Marks
 Math, Linear Algebra, Algebra, Transformations, Vectors, Sets

Click to edit the document details