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**Unformatted text preview: **A brief survey of linear algebra Tom Carter http://astarte.csustan.edu/ tom/linear-algebra Santa Fe Institute Complex Systems Summer School June, 2001 1 Our general topics: } Why linear algebra } Vector spaces (ex) } Examples of vector spaces (ex) } Subspaces (ex) } Linear dependence and independence (ex) } Span of a set of vectors (ex) } Basis for a vector space (ex) } Linear transformations (ex) } Morphisms mono, epi, and iso (ex) } Linear operators (ex) } Normed linear spaces (ex) } Eigenvectors and eigenvalues (ex) } Change of basis (ex) } Trace and determinant (ex) } References (ex): exercises. 2 Why linear algebra Linear models of phenomena are pervasive throughout science. The techniques of linear algebra provide tools which are applicable in a wide variety of contexts. Beyond that, linear algebra courses are often the transition from lower division mathematics courses such as calculus, probability/statistics, and elementary differential equations, which typically focus on specific problem solving techniques, to the more theoretical axiomatic and proof oriented upper division mathematics courses. I am going to stay with a generally abstract, axiomatic presentation of the basics of linear algebra. (But Ill also try to provide some practical advice along the way . . . :-) 3 Vector spaces The first thing we need is a field F of coefficients for our vector space. The most frequently used fields are the real numbers R and the complex numbers C . A field F = ( F , + , * ,- ,- 1 , , 1) is a mathematical object consisting of a set of elements ( F ), together with two binary operations (+ , * ), two unary operations (- ,- 1 ), and two distinguished elements 0 and 1 of F , which satisfy the fundamental properties: 1. F is closed under the four operations: + : F F F * : F F F- : F F- 1 : F F F = { a F | a 6 = 0 } Of course, we usually write a + b,a * b,- a , and a- 1 instead of +( a,b ) , * ( a,b ) ,- ( a ), and- 1 ( a ). 4 2. + and * are commutative and associative, and satisfy the distributive property. That is, for a,b,c F : a + b = b + a a * b = b * a ( a + b ) + c = a + ( b + c ) ( a * b ) * c = a * ( b * c ) a * ( b + c ) = a * b + a * c 3. 0 is the identity element and - is the inverse for addition. 1 is the identity element and a- 1 is the inverse for multiplication. That is, for a F : a + 0 = a a + (- a ) = 0 a * 1 = a a * ( a- 1 ) = 1 ( a 6 = 0) 5 4. Although we wont need it for most of linear algebra, Ill mention that R and C are both complete (Cauchy sequences have limits), and R is fully ordered ( a < b or b < a or a = b for all a,b R ). 5. As needed, we will identify R as a subfield of C , and we will typically write elements of C as a + bi , where a and b are real and i 2 =- 1....

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