Scientific Computing

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CS205 Review Session # 3 Notes Linear Dependence In order to prove that a set of nonzero vectors are linearly independent, we can assume that they are linearly dependent and show that this leads to a contradiction. If a set of vectors V = { v i = 0 | i ∈ { 1 , . . . , m }} is linearly dependent, the following three conditions all hold: 1. c i IR (at least two of which are nonzero) such that m i =1 c i v i = 0 2. v k V such that v k = i = k c i v i = 0 with at least one c i = 0 3. v k V that can be written as a linear combination of a linearly independent subset of V , i.e. v k = i =1 c i v i (with some renumbering), where the choice of { c i } is unique . This last formulation gives, in some sense, the minimal linearly dependent subset of V , since the removal of any vector from { v k , v 1 , . . . , v } yields a linearly independent set. Deflation Matrices Consider a symmetric n × n matrix A where λ is an eigenvalue of A and q the corresponding (normalized) eigenvector. We define the deflation of A with respect to ( λ, q ) to be: D ( λ, q ) = A - λ qq T We note that this matrix has the following properties: 1. q NS( A - λ qq T ). Equivalently, we can say that
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