Page 170 - Matrix Analysis & Applied Linear Algebra

4.1 Spaces and Subspaces 165

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For example, if u = 0 is a vector in , then span {u} is the straight
line passing through the origin and u. If S = {u, v}, where u and v are two
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nonzero vectors in not lying on the same line, then, as shown in Figure 4.1.4,
span (S) is the plane passing through the origin and the points u and v. As we
n
will soon see, all subspaces of are of the type span (S), so it is worthwhile
to introduce the following terminology.

Spanning Sets

• For a set of vectors S = {v 1 , v 2 ,..., v r } , the subspace

span (S)= {α 1 v 1 + α 2 v 2 + ··· + α r v r }

generated by forming all linear combinations of vectors from S is
called the space spanned by S.
• If V is a vector space such that V = span (S) , we say S is a
spanning set for V. In other words, S spans V whenever each
vector in V is a linear combination of vectors from S.

Example 4.1.6

(i) In Figure 4.1.4, S = {u, v} is a spanning set for the indicated plane.

1 2
2
(ii) S = , spans the line y = x in .
1 2
      
1 0 0
 
3
0
1
0
(iii) The unit vectors e 1 =   , e 2 =   , e 3 =   span .
0 0 1
 
n n
(iv) The unit vectors {e 1 , e 2 ,..., e n } in form a spanning set for .
2 n
(v) The finite set 1,x,x ,...,x spans the space of all polynomials such
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that deg p(x) ≤ n, and the infinite set 1,x,x ,... spans the space of all
polynomials.
Example 4.1.7
m×1
Problem: For a set of vectors S = {a 1 , a 2 ,..., a n } from a subspace V⊆ ,
let A be the matrix containing the a i ’s as its columns. Explain why S spans V
if and only if for each b ∈V there corresponds a column x such that Ax = b
(i.e., if and only if Ax = b is a consistent system for every b ∈V).