Page 200 - 35Linear Algebra

200 Subspaces and Spanning Sets

       
1 1 a x
r 1   + r 2  2  + r 3   =   .
y
1
0
a −3 0 z
2
1
3
We can write this as a linear system in the unknowns r , r , r as follows:
1
     
1 1 a r x
0 2 1 r = y .
   2   
a −3 0 r 3 z
 
1 1 a
If the matrix M =  0 2 1  is invertible, then we can find a solution
a −3 0
1
   
x r
M −1   =  2 
r
y
z r 3
 
x
3
y
for any vector   ∈ R .
z
Therefore we should choose a so that M is invertible:
2
i.e., 0 6= det M = −2a + 3 + a = −(2a − 3)(a + 1).
3
3
Then the span is R if and only if a 6= −1, .
2

Linear systems as spanning sets

Some other very important ways of building subspaces are given in the
following examples.

Example 112 (The kernel of a linear map).

Suppose L : U → V is a linear map between vector spaces. Then if

0
L(u) = 0 = L(u ) ,
linearity tells us that

0
0
L(αu + βu ) = αL(u) + βL(u ) = α0 + β0 = 0 .
200