problemset3 - MAS 213 Linear Algebra II Problem list for...

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MAS 213: Linear Algebra II. Problem list for Week #3. Tutorial on 21st September. This week’s topics: A basis for a vector space. Techniques for manipulating bases: Enlarging a linearly independent set to a basis. Reducing a spanning set to a basis. The theory of bases: The replacement theorem. Every basis for a finite dimensional vector space has the same number of elements. Recognizing a basis. Application: Lagrange polynomials. Tutorial problems: Problem 1: (Problems 1.6.3 (d)-(e) in [FIS].) Which of the following sets are bases for R 3 ? 1. { ( - 1 , 3 , 1) , (2 , - 4 , - 3) , ( - 3 , 8 , 2) } 2. { (1 , - 3 , - 2) , ( - 3 , 1 , 3) , ( - 2 , - 10 , - 2) } Problem 2: (Problems 1.6.3 (d)-(e) in [FIS].) Which of the following sets are bases for P 2 ( R )? 1. {- 1 + 2 x + 4 x 2 , 3 - 4 x - 10 x 2 , - 2 - 5 x - 6 x 2 } 2. { 1 + 2 x - x 2 , 4 - 2 x + x 2 , - 1 + 18 x - 9 x 2 } 1
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Problem 3: (Problem 1.6.4 and 1.6.5 in [FIS].) Two quick questions: 1. Is { x 3 - 2 x 2 + 1 , 4 x 2 - x + 3 , 3 x - 2 } a spanning set for P 3 ( R )? 2. Is { (1 , 4 , - 6) , (1 , 5 , 8) , (2 , 1 , 1) , (0 , 1 , 0) } a linearly independent subset of R 3 ?
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