NYC.L5 - MAT-NYC LAB #5 Linear Dependence, Basis and...

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MAT-NYC LAB #5 Linear Dependence, Basis and Dimension 1. Are the following vectors linearly independent? If not, find a dependency relationship. a) [2 , 0 , - 1] , [3 , - 1 , - 4] , [ - 1 , 2 , 1] b) [1 , 1 , 0] , [0 , 1 , - 1] , [ - 2 , 1 , - 3] , [1 , 2 , - 1] 2. Let ~v 1 , ~v 2 , . . . , ~v r be r vectors in R n . Explain how the concept of rank can be used to determine if the vectors are linearly independent. Give any conditions that must be satisfied by r , the number of vectors, in order for the question to be answered positively, then apply your analysis to determine whether the vectors [1 , 2 , 1 , - 3], [ - 1 , 1 , 2 , 2], [1 , 5 , 4 , - 4] and [1 , 8 , 7 , - 4] are linearly independent. 3. Let ~v 1 = [1 , 1 , 1 , 2] ,~v 2 = [1 , - 1 , 2 , 3] ,~v 3 = [0 , - 2 , 1 , 1] ,~v 4 = [1 , - 2 , 3 , 2] and ~v 5 = [0 , - 1 , 1 , - 1]. Find a subset of { ~v 1 ,~v 2 ,~v 3 ,~v 4 ,~v 5 } which is a basis for their linear span W . Write the deleted vector(s) as linear combinations of the basis vectors. What is the dimension of
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NYC.L5 - MAT-NYC LAB #5 Linear Dependence, Basis and...

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