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 premisebut 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 isorthogonal to all of them (that is, for all i = 1,2,...n, (x, v₁) = 0). Then x is orthogonal to anyy € Span(S).4. Let S ={V₁, V2,..., Vn} V be an orthonormal set of vectors in V. Then for all non-zerox € 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 forall 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 thatX=CV1+tCnvn7. 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 thatx = C₁ V₁ + ... + Cn Vn8. 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.

Answered: Image /qna-images/answer/345e3373-28e2-434e-992b-080529f4f295.jpg

1. Triangle inequality for inner products: For all a, b, c € V, (a, c) ≤ (a, b) + (b, c). = 0 and (b, c) =0 then (a, c) = 0. : 2. Transitivity of orthogonality: For all a, b, c € V, if (a, b) 3. Orthogonality closed under addition: Suppose S = {V₁,..., Vn} CV is a set of vectors, and x is orthogonal to all of them (that is, for all i = 1,2,...n, (x, vi) = 0). Then x is orthogonal to any y € Span(S).

1. Triangle inequality for inner products: For all a, b, c € V, (a, c) ≤ (a, b) + (b, c). = 0 and (b, c) =0 then (a, c) = 0. : 2. Transitivity of orthogonality: For all a, b, c € V, if (a, b) 3. Orthogonality closed under addition: Suppose S = {V₁,..., Vn} CV is a set of vectors, and x is orthogonal to all of them (that is, for all i = 1,2,...n, (x, vi) = 0). Then x is orthogonal to any y € Span(S).

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.3: Spanning Sets And Linear Independence

Problem 21EQ

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