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Linear Combinations of Vectors

Module by: Mark A. Davenport. E-mail the author

Suppose we have a set of vectors v1,v2,...,vNv1,v2,...,vN that lie in a vector space VV. Given scalars α1,α2,...,αNα1,α2,...,αN, observe that the linear combination

α 1 v 1 + α 2 v 2 + . . . + α N v N α 1 v 1 + α 2 v 2 + . . . + α N v N
(1)

is also a vector in VV.

Definition 1

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

Example 1: V = R 3 V = R 3

v 1 = 1 1 0 , v 2 = 0 1 0 . v 1 = 1 1 0 , v 2 = 0 1 0 .
(2)

span ({v1,v2})= span ({v1,v2})= the x1x2x1x2-plane, i.e., for any x1,x2x1,x2 we can write x1=α1x1=α1 and x2=α1+α2x2=α1+α2 for some α1,α2Rα1,α2R.

Figure 1: Illustration of the set of all linear combinations of v1v1 and v2v2, i.e., the x1x2x1x2-plane.
An illustration of the set of all linear combinations of v_1 and v_2, i.e., the x_1 x_2 - plane.

Example 2

V={f:f(t) is periodic with period 2π}V={f:f(t) is periodic with period 2π}, M={ejkt}k=-BBM={ejkt}k=-BB

span (M)= span (M)= periodic, bandlimited (to BB) functions, i.e., f(t)f(t) such that f(t)=Bk=-BcKejktf(t)=Bk=-BcKejkt for some c-B,c-B+1,...,c0,c1,...,cBCc-B,c-B+1,...,c0,c1,...,cBC.

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