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Unformatted text preview: MATH 304, Fall 2011 Linear Algebra Homework assignment #6 (due Thursday, October 20) All problems are from Leons book (8th edition). Section 3.5: 1b, 5a, 6a, 6b Section 3.6: 1a, 2a, 2b, 2c, 9, 16 MATH 304 Linear Algebra Lecture 12: Rank and nullity of a matrix. Basis Definition. Let V be a vector space. A linearly independent spanning set for V is called a basis . Equivalently, a subset S V is a basis for V if any vector v V is uniquely represented as a linear combination v = r 1 v 1 + r 2 v 2 + + r k v k , where v 1 , . . . , v k are distinct vectors from S and r 1 , . . . , r k R . Remark on uniqueness. Expansions v = 2 v 1 + 3 v 2 v 3 and v = 2 v 1 + 3 v 2 v 3 + 0 v 4 are considered the same. Dimension Theorem 1 Any vector space has a basis. Theorem 2 If a vector space V has a finite basis, then all bases for V are finite and have the same number of elements. Definition. The dimension of a vector space V , denoted dim V , is the number of elements in any of its bases. Examples. dim R n = n M m , n ( R ): the space of m n matrices; dim M m , n = mn P n : polynomials of degree less than n ; dim P n = n P : the space of all polynomials; dim P = { } : the trivial vector space; dim { } = 0 Theorem Let v 1 , v 2 , . . . , v n be vectors in R n . Then the following conditions are equivalent: (i) { v 1 , v 2 , . . . , v n } is a basis for R n ; (ii) { v 1 , v 2 , . . . , v n } is a spanning set for R n ; (iii) { v 1 , v 2 , . . . , v n } is a linearly independent set. Theorem Let S be a subset of a vector space V . Then the following conditions are equivalent: (i) S is a linearly independent spanning set for V , i.e., a basis; (ii) S is a minimal spanning set for V ; (iii) S is a maximal linearly independent subset of V . How to find a basis? Theorem Let V be a vector space. Then (i) any spanning set for V can be reduced to a minimal spanning set; (ii) any linearly independent subset of V can be extended to a maximal linearly independent set. That is, any spanning set contains a basis, while any linearly independent set is contained in a basis....
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This note was uploaded on 11/08/2011 for the course MATH 304 taught by Professor Hobbs during the Spring '08 term at Texas A&M.
 Spring '08
 HOBBS
 Linear Algebra, Algebra

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