Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question
Here are a collection of conjectures. Which are true, and which are false? • If it is true, provide a formal proof demonstrating so. • If it is false, give a counterexample, clearly stating why your counterexamples satisfifies the premise but not the conclusion.
1. Triangle inequality for inner products: For all a, b, c € V, (a, c) ≤ (a, b) + (b, c). = 2. Transitivity of orthogonality: For all a, b, c € V, if (a, b) = 0 and (b, c) = 0 then (a, c) = 0. 3. Orthogonality closed under addition: Suppose S {V₁,..., Vn} V is a set of vectors, and x is orthogonal to all of them (that is, for all i = 1,2,...n, (x, v₁) = 0). Then x is orthogonal to any y € Span(S). 4. Let S = {V₁, V2,..., Vn} V be an orthonormal set of vectors in V. Then for all non-zero x € V, if for all 1 ≤ i ≤ n we have (x, vi) = 0 then x Span(S). 5. Let S = {V₁, V2, ..., Vn} ≤ V be a set of vectors in V (no assumption of orthonormality). Then for all non-zero x € V, if for all 1 ≤ i ≤ n we have (x, V₁) = 0 then x Span(S). 6. Let S = {V₁,..., Vn} be a set of orthonormal vectors such that Span(S) = V, and let x € V. Then there is a unique set of coefficients c₁,..., Cn such that X=CV1+tCnvn 7. Let S = {V₁,..., Vn} be a set of vectors (no assumption of orthonormality) such that Span(S) = V, and let x € V. Then there is a unique set of coefficients c₁,..., Cn such that x = C₁ V₁ + ... + Cn Vn 8. Let S = {V₁, V2,...) , Vn} CV be a set of vectors. If all the vectors are pairwise linearly independent (i.e., for any 1 ≤ i ‡ j ≤n, then only solution to ciV₁ +CjVj = 0 is the trivial solution c¿ = c; = 0.) then the set S is linearly independent.
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Transcribed Image Text:1. Triangle inequality for inner products: For all a, b, c € V, (a, c) ≤ (a, b) + (b, c). = 2. Transitivity of orthogonality: For all a, b, c € V, if (a, b) = 0 and (b, c) = 0 then (a, c) = 0. 3. Orthogonality closed under addition: Suppose S {V₁,..., Vn} V is a set of vectors, and x is orthogonal to all of them (that is, for all i = 1,2,...n, (x, v₁) = 0). Then x is orthogonal to any y € Span(S). 4. Let S = {V₁, V2,..., Vn} V be an orthonormal set of vectors in V. Then for all non-zero x € V, if for all 1 ≤ i ≤ n we have (x, vi) = 0 then x Span(S). 5. Let S = {V₁, V2, ..., Vn} ≤ V be a set of vectors in V (no assumption of orthonormality). Then for all non-zero x € V, if for all 1 ≤ i ≤ n we have (x, V₁) = 0 then x Span(S). 6. Let S = {V₁,..., Vn} be a set of orthonormal vectors such that Span(S) = V, and let x € V. Then there is a unique set of coefficients c₁,..., Cn such that X=CV1+tCnvn 7. Let S = {V₁,..., Vn} be a set of vectors (no assumption of orthonormality) such that Span(S) = V, and let x € V. Then there is a unique set of coefficients c₁,..., Cn such that x = C₁ V₁ + ... + Cn Vn 8. Let S = {V₁, V2,...) , Vn} CV be a set of vectors. If all the vectors are pairwise linearly independent (i.e., for any 1 ≤ i ‡ j ≤n, then only solution to ciV₁ +CjVj = 0 is the trivial solution c¿ = c; = 0.) then the set S is linearly independent.
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