Abstract Linear Algebra Notes

Abstract Linear Algebra Notes - Abstract Linear Algebra...

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Abstract Linear Algebra Notes 9/26 Definition – Let F be a set with at least two elements. Then is a field if: 1. F is an abelian group under + with additive identity element 0; 2. F\{0} is an abelian group under ; ∙ with multiplicative identity 1 3. Distribution laws hold: Definition – G is a group under if: 1. G is closed under : 2. Has identity element, e 3. Associative laws hold 4. Inverses: Examples: R – field of real numbers (under usual addition and multiplication) C – field of complex numbers (“ “) Q – field of rational numbers (“ “) F p – field of integers modulo p where p is a prime Definition - characteristic of F “char(F)” is the least positive integer n s.t. If such n does not exist, then the characteristic of F is 0. Notice: char(R)=char(C)=char(Q)=0, char(F p )=p Proposition – The characteristic of a field can’t be composite. Pf: Suppose that the characteristic of a field is a composite number n. So, n=a b, where 1 < a,b < n and a,b F. Notice, | , since a < n, b < n, and n is the characteristic of the field. Therefore, n must be prime QED Definition – A vector space (linear space) over a field F consists of the following: 1. A field F of scalars; 2. A set V of objects (V ), called vectors; 3. A rule (or operation), called vector addition, +: V x V , V where V is ; an abelian group under addition
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4. A rule (or operation), called scalar multiplication, * : F x V V, where: (a) (identity) (b) (associative law) (c) (dist. law). Examples of vector spaces over a field F (more info view Hoffman pgs 29-31): 1. The n-tuple space, F n 2. The space of matrices, 3. The space of functions from a space to a field 4. The space of polynomial functions over a field F 5. Let V be a vector space over F and let S be a nonempty set. Let = set of all functions S to V. Propositions – Let V be a vector space over F. (1) Then the additive identity, 0, is unique. Pf: Suppose there is two 0 1 and 0 2 . Then, by using the definition of additive identity we have shown that the additive identity is unique. QED (2) If there exists a pair s.t. , the w is the additive identity. Pf: Take any and consider Therefore, w is the additive identity element. QED
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Abstract Linear Algebra Notes - Abstract Linear Algebra...

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