About Conjectures

a If it is true, provide a formal proof demonstrating so. u If it is false, give a counterexample, clearly stating why your counterexamples satises the premise but not the conclusion. (No marks for just starting True/False.) Hint: There's quite a few questions here, but each is relatively simple (the counterexamples aren't very complicated, and the proofs are short.) Try playing around with a few examples rst to get an intuitive feeling if the statement is true before trying to prove it. Let V be a vector space, and let (-, -) : V x V ) 1R be an inner product over V. 1. 2. rTriangle inequality for inner products: For all a, b, c E V. (a,c) g (a, b) + (b, c). Transitivity of orthogonality: For all a, b, c E V, if (a, b) : 0 and (b,c) : 0 then (a, c) : O. . Orthogonality closed under addition: Suppose S 2 {V1, . . . , v,,} Q V is a set of vectors, and x is orthogonal to all of them (that is, for all i = 1, 2, . . . n, (xvi) : 0). Then x is orthogonal to any y E Span(S]. Let S = {V1,V2, . . .,v,,} Q V be an orthonormal set of vectors in V. Then for all nonzero x E V, if for all 1 S i S n we have (x,vi) = 0 then x E Span(S). . Let S : {V1, V2, . . . , v.,,,} C; V be a set of vectors in V (no assumption of orthonormality). Then for all non-zero x E V, if for all 1 S 3' S n we have (x, vi) = 0 then x E Span(S). . Let S : {V1, . . . ,v,,} be a set of orthonormal vectors such that Span(S) : V, and let x E V. Then there is a unique set of coefcients c1, . . . ._ on such that x : clvl + ...+c.,,v.,, . Let S 2 {V1, . . . , v,,} be a set of vectors (no assumption of orthonormality) such that Span(S) = V, and let x E V. Then there is a unique set of coefcients 01,...,c.,, such that x = clvl + ...+c.nv.,, . Let S : {V1, v2, . . . ,v,,} Q V be a set of vectors. If all the vectors are pairwise linearly independent (i.e.. for any 1 S i aj S n, then only solution to civi + cjvj : 0 is the trivial solution c, : cj : 0.) then the set S is linearly independent