Annihilation By a Set of Linear Functionals

  be a vector space over a field  
. The annihilator of a subspace  
  is the set  
  of all linear functionals  
\[f: V \rightarrow F\]
  such that  
\[f( \alpha)=0\]
  for all  
\[\alpha \in W\]
Obviously the zero function is in  
Also, if  
\[\alpha , \; \beta \in W^0, \; a, \; b \in F\]
\[f( a \alpha + b \beta ) = a f( \alpha ) + b f( \beta )=0 \rightarrow a \alpha + b \beta \in W^0\]

  is closed under addition and scalar multiplication, so the set  
  is a subspace of  
, the space dual to  
To find the annihilator for
\[f_1(x_1, \; x_2, \; x_3, \; x_4)=x_1+2x_2+2x_3+x_4\]

\[f_2(x_1, \; x_2, \; x_3, \; x_4)=2x_2+x_4\]

\[f_1(x_1, \; x_2, \; x_3, \; x_4)=-2x_1-4x_3+3x_4\]

We solve the simultaneous equations
(1)-(2) gives the system
(6)+2(1) gives the system
  then from (5)  
. In (1) put  
  so that vectors of the form  
  are annihilated by the functionals above.

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