Page 198 - 35Linear Algebra

198 Subspaces and Spanning Sets

3
In this case, the vectors in U define the xy-plane in R . We can view the
xy-plane as the set of all vectors that arise as a linear combination of the two
vectors in U. We call this set of all linear combinations the span of U:
  
  
1 0
 

0
1
span(U) = x   + y   x, y ∈ R .

0 0
 
Notice that any vector in the xy-plane is of the form
     
x 1 0
y = x 0 + y 1 ∈ span(U).
     
0 0 0
Definition Let V be a vector space and S = {s 1 , s 2 , . . .} ⊂ V a subset of V .
Then the span of S, denoted span(S), is the set

1
i
N
2
span(S) := {r s 1 + r s 2 + · · · + r s N | r ∈ R, N ∈ N}.
1
That is, the span of S is the set of all finite linear combinations of
elements of S. Any finite sum of the form “a constant times s 1 plus a constant
2
times s 2 plus a constant times s 3 and so on” is in the span of S. .
 
0
3
1 , and set
Example 110 Let V = R and X ⊂ V be the x-axis. Let P =  
0
S = X ∪ {P} .

       
2 2 2 0
3
The vector   is in span(S), because   =   +3   . Similarly, the vector
3
1
0
0 0 0 0
       
−12 −12 −12 0
17.5 is in span(S), because 17.5 = 0 +17.5 1 . Similarly, any vector
       
0 0 0 0
1
Usually our vector spaces are defined over R, but in general we can have vector spaces
i
defined over different base fields such as C or Z 2 . The coefficients r should come from
whatever our base field is (usually R).
2
It is important that we only allow finitely many terms in our linear combinations; in
the definition above, N must be a finite number. It can be any finite number, but it must
be finite. We can relax the requirement that S = {s 1 , s 2 , . . .} and just let S be any set of
2
1
i
N
vectors. Then we shall write span(S) := {r s 1 +r s 2 +· · ·+r s N | r ∈ R, s i ∈ S, N ∈ N, }
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