Question Solved1 Answer 9) The convex hull of a set of vectors xi , i = 1, . . . , n is the set of all vectors of the form x = X n i=1 αixi , where αi ≥ 0 and P i αi = 1. Given two sets of vectors, show that either they are linearly separable or their The convex \( h u l l \) of a set of vectors \( \boldsymbol{x}_{i}, i=1, \ldots, n \) is the set of all vectors of the form \[ \boldsymbol{x}=\sum_{i=1}^{n} \alpha_{i} \boldsymbol{x}_{i} \] where \( \alpha_{i} \geq 0 \) and \( \sum_{i} \alpha_{i}=1 \). Given two sets of vectors, show that either they are linearly separable or their convex hulls intersect. (To answer this, suppose that both statements are true, and consider the classification of a point in the intersection of the convex hulls.)

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9) The convex hull of a set of vectors xi , i = 1, . . . , n is the set of all vectors of the form x = X n i=1 αixi , where αi ≥ 0 and P i αi = 1. Given two sets of vectors, show that either they are linearly separable or their convex hulls intersect. (To answer this, suppose that both statements are true, and consider the classification of a point in the intersection of the convex hulls.)

Transcribed Image Text: The convex \( h u l l \) of a set of vectors \( \boldsymbol{x}_{i}, i=1, \ldots, n \) is the set of all vectors of the form \[ \boldsymbol{x}=\sum_{i=1}^{n} \alpha_{i} \boldsymbol{x}_{i} \] where \( \alpha_{i} \geq 0 \) and \( \sum_{i} \alpha_{i}=1 \). Given two sets of vectors, show that either they are linearly separable or their convex hulls intersect. (To answer this, suppose that both statements are true, and consider the classification of a point in the intersection of the convex hulls.)
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Transcribed Image Text: The convex \( h u l l \) of a set of vectors \( \boldsymbol{x}_{i}, i=1, \ldots, n \) is the set of all vectors of the form \[ \boldsymbol{x}=\sum_{i=1}^{n} \alpha_{i} \boldsymbol{x}_{i} \] where \( \alpha_{i} \geq 0 \) and \( \sum_{i} \alpha_{i}=1 \). Given two sets of vectors, show that either they are linearly separable or their convex hulls intersect. (To answer this, suppose that both statements are true, and consider the classification of a point in the intersection of the convex hulls.)
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Consider a set of foints {z^(m)y:} and it 1 correaponding comvex hull. The two sets of the points be lixealy separable if there exists a veelor hat(omega) and a sealor wo suin that{:[, hat(omega)^(T)x^(n)+w_(0) > 0,AAx_(n)],[" s ", hat(omega)^(T)z_(m)^(m)+w_(0) < 0,AAz_(m)]:}Now we show that if their convex hulls intersest, the two sete of points cannot be livearly separable, and convercely that if they ave lixealy separable, ther convex hulls to notSolnt- Pirstlet calculate the lixeor discriminent for the pointe belonging to tue two convex hall. For the conwer hall of {x^(n)] the tineor. diserimivent isy(x)= hat(w)^(pi)x^(n)+w_(0)-(1)Again ne know x=sumalpha_(n)x^(n) whene x_(n)⩾0 &amp; &amp; s ... See the full answer