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1.8 Linear combinations of vectors

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Suppose we have a set of vectors v 1 , v 2 , . . . , v N that lie in a vector space V . Given scalars α 1 , α 2 , . . . , α N , observe that the linear combination

α 1 v 1 + α 2 v 2 + . . . + α N v N

is also a vector in V .

Definition 1

Let M ⊂ V be a set of vectors in V . The span of M , written span ( M ) , is the set of all linear combinations of the vectors in M .

V = R 3

v 1 = 1 1 0 , v 2 = 0 1 0 .

span ( { v 1 , v 2 } ) = the x 1 x 2 -plane, i.e., for any x 1 , x 2 we can write x 1 = α 1 and x 2 = α 1 + α 2 for some α 1 , α 2 ∈ R .

An illustration of the set of all linear combinations of v_1 and v_2, i.e., the x_1 x_2 - plane.
Illustration of the set of all linear combinations of v 1 and v 2 , i.e., the x 1 x 2 -plane.

V = { f : f ( t ) is periodic with period 2 π } , M = { e j k t } k = - B B

span ( M ) = periodic, bandlimited (to B ) functions, i.e., f ( t ) such that f ( t ) = ∑ B k = - B c K e j k t for some c - B , c - B + 1 , . . . , c 0 , c 1 , . . . , c B ∈ C .
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Source:  OpenStax, Digital signal processing. OpenStax CNX. Dec 16, 2011 Download for free at http://cnx.org/content/col11172/1.4
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