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lec3-1 - More chapter 3.linear dependence and independence...

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More chapter 3...linear dependence and independence... vectors It is important to determine if a set of vectors is linearly dependent or independent Consider a set of vectors ~ A , ~ B , and ~ C . If we can find a , b , and c such that, a ~ A + b ~ B + c ~ C = 0 Then this set of vectors is linearly dependent. More generally, if we have N vectors v n , and we can find a set of coefficients c n , N X n =1 c n v n = 0 Then the vectors are linearly dependent. (Note that we don’t count the case where c n = 0 for all n). Patrick K. Schelling Introduction to Theoretical Methods
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Linear dependence and independence continued, and homogeneous equations For example, think of vectors ~ A , ~ B , and ~ C in 3 dimensions that all lie in the same plane. Since we only need two vectors to define a plane, these vectors must be linearly dependent. We can take the condition N n =1 c n v n = 0 and write a matrix A whose columns are the vectors v n , and a column vector c whose elements are the c n , Then we solve the homogeneous equations Ac = 0 Patrick K. Schelling Introduction to Theoretical Methods
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Homogeneous equations, continued Trivial solution c = 0 always exists If we have n equations and n unknowns, only a non-trivial solution exists if the number of linearly independent equations (rows) is less than the number of unknowns Think of row reduction; at least one row must be reduced to zero In elementary row reduction, combining rows does not change determinant If we have n equations and n unknowns, a non-trivial solution exists only if detA = 0 If we have more vectors than the dimension of the vectors, they are always dependent Patrick K. Schelling Introduction to Theoretical Methods
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Connection to Cramer’s rule a 1 x + b 1 y = c 1 a 2 x + b 2 y = c 2 Using row-reduction, we find x = c 1 b 1 c 2 b 2 a 1 b 1 a 2 b 2 y = a 1 c 1 a 2 c 2 a 1 b 1 a 2 b 2 Thus x = 0, y = 0 if c 1 = c 2 = 0, unless det A = 0 Patrick K. Schelling Introduction to Theoretical Methods
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Eigenvalues and eigenvectors; diagonalization
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