2009MT2

# 2009MT2 - Second Midterm MAT26500 11,12 Thursday Answer as...

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Unformatted text preview: Second Midterm MAT26500 - 11,12 Thursday November 12, 2009 Answer as many of the following problems as possible. No tools other than a pen/pencil are permitted. You have the entire class period to work. Show all work prominently , highlighting key conclusions and explicitly answering the problem stated. There is a maximum score of 100 points. Problem 1 10 points For each of the following vector spaces and sets of vec- tors, answer the following questions: 1. Does the set span the vector space V ? 2. Is the set linearly independent? 3. Is the set a basis? Calculate, or give appropriate reason (I will assume you know the dimensions of the vector spaces given). R 2 : braceleftbiggbracketleftbigg 1 7 bracketrightbigg , bracketleftbigg 2 14 bracketrightbigg , bracketleftbigg 1 1 bracketrightbiggbracerightbigg R 3 : 1 7 1 , 1 2 2 R 4 : 1 2 , 1 1 1 , 1 3 1 1 R 3 : 1 1 , 1 1 , 1 1 P 2 [ x ] : braceleftbig t + 1 ,t- 1 ,- t 2 + t + 1 bracerightbig Answer: 1. Spans, not linearly independent, therefore not a basis. 2. Doesn’t span (therefore not a basis), linearly independent. 3. Doesn’t span, not linearly independent, therefore not a basis. 4. Spans, linearly independent, therefore a basis. 5. Spans, linearly independent, therefore a basis. (It is easy to write 1 ,t,t 2 as linear combinations of these vectors, therefore they span, and since P 2 [ x ] has dimension 3, it is also linearly independent and therefore a basis). 1 Problem 2 25 points Consider the matrices A = 1 3 14- 2 14- 2- 5- 25 7- 33 5 8 49- 30 103- 2- 3- 19 9- 35 , B = 1 5 7 1 3 1 1- 2 Assume in the following questions that the matrix B is obtained from the matrix A by row reduction (because it is!). 1. Find bases for null( A ) , row( A ) , col( A ) . Ans: For null( A ) , row( A ) use the matrix B to find the bases: we find for the null space the basis { [- 7- 1 2 1] T , [- 5- 3 1 0] T } for the row space the basis (pardon the bad spacing; these are just the non-zero rows of B ): [1 , , 5 , , 7] , [0 , 1 , 3 , , 1] , [0 , , , 1 ,- 2] for the column space, first observe that the 1st, 2nd, 4th columns are the columns of B containing the ”pivots” (leading 1 s), so take the 1 , 2 , 4 th columns of A as the basis: [1 ,- 2 , 5 ,- 2] T , [3 ,- 5 , 8 ,- 3] T , [- 2 , 7 ,- 30 , 9] T 2. What does the dimension theorem claim about the matrix A . Use your answer in the previous part to verify....
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2009MT2 - Second Midterm MAT26500 11,12 Thursday Answer as...

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