Discrete Mathematical Structures by Pearson

12-14 Discrete Mathematical Structures

We observe that the sum of any two linear combinations of

 

xx x

m12

,,..., is another linear

combination of these vectors and so also is a acxc

xc

x

mm

( ... ),

11 22

 

+++

a ∈ F.

Hence, the set U of all linear combination of vectors

 

xx x

m12

,,...,

in a vector space V, is a

subspace of V.

We also say that the subspace U is spanned or generated by vectors

 

xx x

m12

,,..., .

If S denotes the set of vectors

 

xx x

m12

,,..., . Then U is also called Linear span of S and is

denoted L(s). Thus,

Ls cx cx cx cF

xV

mm

ii

()

{)

:,,

=+++ ∈∈

11 22







m a positive integer}

Example 1 Determine the subspace spanned by

S = {(2, 3, 1), (1, 0, 2)} in R

3

The required vector space U is given by,

U = {a (2, 3, 1) + b(1, 0, 2), a, b