WI08Notes-Jan16-lincomb

WI08Notes-Jan16-lincomb - v k U . The ane 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 .T h e 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 .T h e 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 =1and 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 ,
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Unformatted text preview: v k U . The ane subspace generated by v 1 , v 2 , . . . , v k is the set of all ane 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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This note was uploaded on 09/25/2010 for the course MATH 1bPRAC taught by Professor Wilson during the Winter '09 term at Caltech.

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