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WI08Notes-Jan16-lincomb

WI08Notes-Jan16-lincomb - v k ∈ U The affine subspace...

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Math 1b — Types of linear combinations A vector u is a linear combination of vectors v 1 , v 2 , . . . , v k when there are scalars c 1 , c 2 , . . . , c k so that u = c 1 v 1 + c 2 v 2 + . . . + c k v k . A nonempty set U of vectors is a subspace (or linear subspace ) when U contains all linear combinations of vectors v 1 , v 2 , . . . , v k U . The span of v 1 , v 2 , . . . , v k is the set of all linear combinations of v 1 , v 2 , . . . , v k . A vector u is a nonnegative linear combination of vectors v 1 , v 2 , . . . , v k when there are nonnegative scalars c 1 , c 2 , . . . , c k so that u = c 1 v 1 + c 2 v 2 + . . . + c k v k . A nonempty set U of vectors is a convex cone when U contains all nonegative linear combinations of vectors v 1 , v 2 , . . . , v k U . The convex cone generated by v 1 , v 2 , . . . , v k is the set of all nonnegative linear combinations of v 1 , v 2 , . . . , v k . A vector u is an affine combination of vectors v 1 , v 2 , . . . , v k when there are scalars c 1 , c 2 , . . . , c k so that c 1 + c 2 + . . . + c k = 1 and u = c 1 v 1 + c 2 v 2 + . . . + c k v k . A nonempty set U of vectors is an affine subspace when U contains all affine combinations of vectors v 1 , v 2 , . . . ,
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Unformatted text preview: v k ∈ U . The affine subspace generated by v 1 , v 2 , . . . , v k is the set of all affine combinations of v 1 , v 2 , . . . , v k . A vector u is an convex combination of vectors v 1 , v 2 , . . . , v k when there are nonneg-ative scalars c 1 , c 2 , . . . , c k so that c 1 + c 2 + . . . + c k = 1 and u = c 1 v 1 + c 2 v 2 + . . . + c k v k . The convex hull of v 1 , v 2 , . . . , v k is the set of all convex combinations of v 1 , v 2 , . . . , v k . A nonempty set U of vectors is an convex set when U contains all convex combinations of vectrors v 1 , v 2 , . . . , v k ∈ U . The parallelopiped spanned by v 1 , v 2 , . . . , v k is the set of all linear combinations u = c 1 v 1 + c 2 v 2 + . . . + c k v k where 0 ≤ c i ≤ 1 for all i = 1 , 2 , . . . , k ....
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