Sec. 3.2 Subspaces (continued)
In the previous example, we see that
N ( A)   v1   v2 ,  and  are arbitrary real numbers
where v1 , v2 are two vectors. This subspace
 v1   v2 ,  and  are arbitrary real numbers
consists of vectors which are linear combinations of v1 , v2 and are called the span of vectors v1 , v2 .
Definition Let v1, v2, . . . , vn be vectors in a vector space V. A sum of the form
α1v1+α2v2+· · ·+αnvn, where α1, . . . , αn are scalars, is called a linear combination
of v1, v2, . . . , vn. The set of all linear combinations of v1, v2, . . . , vn is called
the span of v1, . . . , vn. The span of v1, . . . , vn will be denoted by Span(v1, . . . , vn).
Example 6
In R3, let
1 
 0
 
 
e1   0  , e2  1  ,
 0
 0
 
 
then
  

 

span  e1 , e2      ,  and  are arbitrary real numbers 
 0 

 

Geometric interpretation of span(e1,e2): it is the x-y place of R3.
Question: what is span(e1)?
Obviously span(e1,e2) and span(e1) are subspaces of R3.
In general, if v1, v2, . . . , vn are elements of a vector space V, then Span(v1, v2, . . . , vn)
is a subspace of V.
Definition: Spanning set
Suppose V is a vector space, and v1 , v2 ,
vn V . Then the set {v1, . . . , vn} is a spanning set
for V if V=span(v1,v2,…,vn).
Example: {e1,e2} is a spanning set for R2.
1 
0
Example: If v1    ,
 1
v2    , , then is (v1, v2) a spanning set for R2?
 1
We must determine for any vector [a,b]^T in R2, whether it is possible to find constants α1, α2 such
that
a
1v1   2 v2    ,
b
 
i.e.
1 
 1  a 
 
   
1     2      .
0
1
b
These equations can be written out explicitly as
1   2  a ,
 2  b.
Thus we find
1  a  b,  2  b, so this is possible. As a result, {e1,e2} is a spanning set for R2.
EXAMPLE Which of the following are spanning sets for R3?
(a) {e1, e2, e3, (1, 2, 3)T }
(b) {(1, 1, 1)T , (1, 1, 0)T , (1, 0, 0)T }
(c) {(1, 0, 1)T , (0, 1, 0)T }
HW: Sec. 3.2, 11, 12, 13, 14.