Given a collection x1, x2, …, xk of vectors in ℝn, we say that the vectors form a linearly dependent set if one of them can be written as a linear combination of the others. That is, the set is linearly dependent if there is an index j in the range 1 to k such that
where the ci’s are given numbers.
A set A in ℝn is convex if for any two points x and y in A, the points x + t(y − x), t ∈ [0, 1], all lie within A.
(a) Prove that the open ball A ={x ∈ ℝn | ||x|| < 1} is convex.
(b) Prove that the sphere S = {||x|| ∈ ℝn | ||x|| = 1} is not convex.
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