MT2 - Second Midterm Problems MAT26500 11,12 Here is a...

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Second Midterm Problems MAT26500 - 11,12 Here is a collection of problems to prepare for the midterm. I suggest you use the textbook to supplement your study. In particular, you should answer all the True/False questions at the end of each chapter (at least the ones that apply to the sections we covered, which is nearly all of them), and perhaps some of the Supplementary Exercises. The supplementary exercises seem to be more difficult than the homework/review problems below, so don’t be discouraged if you have some trouble with them. If you plan to do some, here are a few that I would point out (of course you are free and encouraged to try others as well): 1. Chapter 4 Supplementary exercises: 2, 4 , 6, 14, 19a, 19b, 23 2. Chapter 5 Supplementary exercises: 6,9, 11, 14, 17, 18, The following problems probably represent the exam questions better. Problem 1. Which of the following subsets of the given vector space is a sub- space? Give a proof or disprove by demonstrating a property of subspaces which it fails. 1. The first quadrant in R 2 , namely { ( x, y ) | x 0 , y 0 } 2. The x -axis in R 2 , namely { ( x, y ) | x R , y = 0 } = { ( x, 0) | x R } 3. The set of polynomials of degree exactly equal to n , i.e. a n x 2 + a n - 1 x n - 1 + · · · + a 1 x 1 + a 0 x 0 , a n 6 = 0 4. The set of polynomials of degree less than or equal to n , i.e. a n x 2 + a n - 1 x n - 1 + · · · + a 1 x 1 + a 0 x 0 5. The set of all vectors 3 × 1 vectors (interpreted as points in the vector space R 3 ) which satisfy the linear system 2 x - 7 y +3 z = 2 3 x - 2 y - 2 z = - 1 6. The set of all 3 × 1 vectors (interpreted as points in the vector space R 3 ) which satisfy the linear system 2 x - 7 y +3 z = 0 3 x - 2 y - 2 z = 0 1
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Problem 2. For each of the following vector spaces and sets of vectors, answer the following questions: 1. Does the set span the vector space V ? 2. Is the set linearly independent? If not, find a subset that spans the same subspace but is linearly independent. 3. Is the set a basis? 1 7 1 , 1 2 2 , 1 7 , 2 14 1 1 , 1 2 0 0 , 0 1 1 1 1 3 1 1
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