HW1 - Math 104 Fall 2009 Homework 1 Due Wednesday Please...

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Math 104 Fall 2009 Homework 1 Due Wednesday, September 30, 2009 Please write neat and complete solutions to the problem sets. “Neat” means well structured, both esthet- ically and logically. “Complete” means that the grader will need to see a sufficient amount of explanations and details to give you full credit, even if the question only asks for a numerical answer. Thanks. 1. Consider the three vectors v 1 = 1 0 - 1 , v 2 = - 1 1 0 , v 3 = 0 - 1 1 . (a) Find the dimension of the vector space generated by v 1 , v 2 , v 3 . (b) Do v 1 , v 2 , v 3 form a basis for the space they generate? (c) Find a matrix, whose entries are not all zeros, and whose nullspace contains all three vectors v 1 , v 2 , v 3 . (d) Find a vector w such that { v 1 , v 2 , w } is a basis for R 3 . 2. Justify the statement: if a collection of vectors contains the zero vector, then there is no chance that the vectors in the collection be linearly independent.
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Unformatted text preview: 3. Two vectors are said to be collinear when one can be written as a scalar multiple of the other. Consider two vectors u and v that are not collinear. Consider a vector w that does not belong to the linear span of u and v . Prove that u,v,w are linearly independent. 4. A matrix is said to be upper triangular if a ij = 0 for i > j . Consider a generic 3-by-3 upper triangular matrix A = a 11 a 12 a 13 a 22 a 23 a 33 . (a) If a 11 ,a 22 , and a 33 are nonzero, show that the only solution to Ax = 0 is x = 0. (b) If either a 11 = 0, or a 22 = 0 or a 33 = 0, then prove that the columns are linearly dependent. (Consider all three cases separately.) (c) If a 22 = 0, find a nonzero element in the nullspace of A . 1...
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