Let V = (v1,..., Vn) be a family of vectors in R Furthermore, (,) denotes that...

 

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Let V = (v1,..., Vn) be a family of vectors in R Furthermore, (,) denotes that canonical dot product. Then the elements gij of the Gram's matrix G are by means Bij = (vi, vj) Are defined. The determinant of this matrix is called Gram's determinant. a) Assume that the vectors vi are pairwise orthogonal and calculate for this situation the Gram's determinant. b) show that the family of vectors V is linearly independent if and only if the Gram's determinant does not vanish. c) show that for a linearly independent family of vectors V is the Gram matrix is positive definite. A matrix A = M(n x n; C) is called positive definite if for all vectors " xe ch , xe it holds that x Ax > 0. Hint: first show that for n = 2 the condition of positive definiteness of G(v1, v2) corresponds to the Cauchy-Schwarz inequality. Let V = (v1,..., Vn) be a family of vectors in R Furthermore, (,) denotes that canonical dot product. Then the elements gij of the Gram's matrix G are by means Bij = (vi, vj) Are defined. The determinant of this matrix is called Gram's determinant. a) Assume that the vectors vi are pairwise orthogonal and calculate for this situation the Gram's determinant. b) show that the family of vectors V is linearly independent if and only if the Gram's determinant does not vanish. c) show that for a linearly independent family of vectors V is the Gram matrix is positive definite. A matrix A = M(n x n; C) is called positive definite if for all vectors " xe ch , xe it holds that x Ax > 0. Hint: first show that for n = 2 the condition of positive definiteness of G(v1, v2) corresponds to the Cauchy-Schwarz inequality.

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a If the vectors vi are pairwise orthogonal then vi vj 0 for i j The Gram matrix G is then a diagona... View the full answer