problems_2.1.pdf - Math 307 Problems for section 2.1 0 0 1...

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Math 307: Problems for section 2.1 October 27, 2017 1. Are the vectors 1 2 1 2 1 , 1 0 - 2 1 1 , 1 - 1 3 - 2 0 , 0 0 - 2 0 1 , 0 4 - 9 7 3 linearly independent? You may use MAT- LAB/Octave to perform calculations, but explain your answer. 2. Which of the following sets are subspaces of the vector space V ? Why, or why not? (a) The set S = { ( b 1 , b 2 , b 3 ) : b 1 = 0; b 2 , b 3 R } . ( V = R 3 ) (b) The set S = { ( b 1 , b 2 , b 3 ) : b 1 b 2 = 0 , b 3 R } . This is union of the plane b 1 = 0 and the plane b 2 = 0 . ( V = R 3 ) (c) All infinite sequences ( x 1 , x 2 , . . . ) , with x i R and x j = 0 from some fixed point onwards. ( V = R ) (d) All non-increasing sequences ( x 1 , x 2 , . . . ) , with x i R and x j +1 x j for each j . ( V = R ) (e) The set of all polynomial functions, p ( x ) , where p ( x ) = 0 or p ( x ) has degree n for some fixed n 1 . ( V is the vector space of all polynomials.)

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