Lecture Slides 20

Lecture Slides 20 - AMS 210: Applied Linear Algebra...

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AMS 210: Applied Linear Algebra November 24, 2009 AMS 210
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Topics Today Problem Set 10 Definitions: null space, range, vector space Column Space Linear Dependence and Independence Determining Linear Dependence Linear Bases Rank-Nullity Theorem Row Space AMS 210
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Problem Set 10 Due Thursday, December 3. Section 5.1: 5ab, 6ab, 8, 11ab, 18ab, 21, 23c; section 5.2: 6ce, 9bd, 12, 21, 25. Show work; use staples (grader may deduct points otherwise. This means that I shouldn’t get anyone asking in class on December 3 whether I have a stapler. I probably won’t). AMS 210
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Definitions Null space (aka kernel): Null( A ) of a matrix A is the set of vectors x that are solutions to the system of equations Ax = 0. This is a vector space. Range: Range( A ) is the set of vectors b such that Ax = b has a solution. This is also a vector space. Vector space: any set V of vectors such that if x 1 , x 2 are in V , then any linear combination r x 1 + s x 2 is also in V . The range and null space are both vector spaces. AMS 210
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Theorem 1 on Null Spaces Let A be any m -by- n matrix. i If Null( A ) contains one nonzero vector x 0 , then Null( A ) contains an infinite number of vectors; in particular, any multiple r x 0 is in Null( A ). ii If x 0 is in Null( A ) and x * is a solution to Ax = b , then x 0 + x * is also a solution to Ax = b . iii If x 1 , x 2 are two different solutions to Ax = b , for some given b , then their difference x 1 - x 2 is a vector in Null( A ). iv Given a solution x * to Ax = b , then any other solution x q to this matrix equation can be written as x q = x * + x 0 for some x 0 in Null( A ).
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Lecture Slides 20 - AMS 210: Applied Linear Algebra...

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