lecture_19 - MA 265 LECTURE NOTES: FRIDAY, FEBRUARY 22 Span...

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Unformatted text preview: MA 265 LECTURE NOTES: FRIDAY, FEBRUARY 22 Span Linear Combinations. Let ( V, + , ) be a real vector space. Say that S = { v 1 , v 2 , ..., v k } is a subset of vectors from V . Recall that a vector v V is called a linear combination if there exist scalars a 1 , a 2 , ..., a k R such that v = a 1 v 1 + a 2 v 2 + + a k v k = k X i =1 a i v i . The set of all such linear combinations is called the span of S : span S = ( v V v = k X i =1 a i v i for some a i R ) . Example. Let V = M 23 denote the collection of 2 3 matrices, and consider the set S = 1 0 0 0 0 0 , 0 1 0 0 0 0 , 0 0 0 0 1 0 , 0 0 0 0 0 1 . An arbitrary linear combination is in the form v = a 1 0 0 0 0 0 + b 0 1 0 0 0 0 + c 0 0 0 0 1 0 + d 0 0 0 0 0 1 = a b c d . Hence we find that span S = a b c d M 23 a, b, c, d R . Span as Subspace. Let S be a set of vectors in a real vector space V . Then span S is a subspace of V ....
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lecture_19 - MA 265 LECTURE NOTES: FRIDAY, FEBRUARY 22 Span...

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