07-2 - 18.06 QUIZ 1 March 05, 2007 Your PRINTED name is:...

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18.06 QUIZ 1 March 05, 2007 Grading 1 2 3 4 5 Total: Your PRINTED name is: SOLUTIONS Please circle your recitation: (1) M 2 2-131 A. Osorno (2) M 3 2-131 A. Osorno (3) M 3 2-132 A. Pissarra Pires (4) T 11 2-132 K. Meszaros (5) T 12 2-132 K. Meszaros (6) T 1 2-132 Jerin Gu (7) T 2 2-132 Jerin Gu
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Problem 1 (20 points) Are the following sets of vectors in R 3 vector subspaces? Explain your answer. (a) vectors ( x,y,z ) T such that 2 x - 2 y + z = 0 YES NO It is given by a linear equation equal to 0. You can also think about it as the nullspace of the matrix 2 - 2 1 · . (b) vectors ( x,y,z ) T such that x 2 - y 2 + z = 0 YES NO The vector 1 0 - 1 is in the set, but if you multiply by - 1 , - 1 0 1 is not. (c) vectors ( x,y,z ) T such that 2 x - 2 y + z = 1 YES NO It is given by a linear equation not set equal to 0. In particular, it doesn't contain the 0 vector. (d) vectors ( x,y,z ) T such that x = y AND x = 2 z YES NO It is the intersection of two planes! We can think about this set as the nullspace of the matrix 1 - 1 0 1 0 - 2 . (e) vectors ( x,y,z ) T such that x = y OR x = 2 z YES NO It is the union of two planes! Take for example 1 1 0 + 2 0 1 = 3 1 1 which is not in the set.
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Problem 2 (20 points) Let A be a 4 × 3 matrix with linearly independent columns. (a) What are the dimensions of the four fundamental subspaces
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07-2 - 18.06 QUIZ 1 March 05, 2007 Your PRINTED name is:...

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