Page 403 - 35Linear Algebra

G.8 Subspaces and Spanning Sets 403

G.8 Subspaces and Spanning Sets

Linear systems as spanning sets

j
1
n
2
Suppose that we were given a set of linear equations l (x , x , . . . , x ) and we
j
j
j
want to find out if l (X) = v for all j for some vector V = (v ). We know that
we can express this as the matrix equation
X j i j
l x = v
i
i
j i j
where l is the coefficient of the variable x in the equation l . However, this
i
j
is also stating that V is in the span of the vectors {L i } i where L i = (l ) j . For
i
example, consider the set of equations
2x + 3y − z = 5
−x + 3y + z = 1
x + y − 2z = 3
which corresponds to the matrix equation

     
2 3 −1 x 5
 −1 3 1   y  =  1  .
1 1 −2 z 3

We can thus express this problem as determining if the vector

 
5
V =  1 
3

lies in the span of

    
2 3
 −1 
, , .
 −1   3   1 
1 1 −2
 
Hint for Review Problem 2

3
For the first part, try drawing an example in R :

403