Linear Algebra--Eq. for subspace.

Find an Ax + By + Cz  = D Equation for the subspace of  spanned by :

 

WE GET FROM THE ABOVE MATRIX EQN.
X=A+B...........1
Y=A+2B........2
Z=A+4B................3
EQN.2 - EQN. 1
Y-X=B..............4
PUTTING IN EQN.1 , WE GET
X=A+Y-X
A=2X-Y..........................5
PUTTING EQNS. 4 AND 5 IN EQN.3
Z=2X-Y+4(Y-X)=2X-Y+4Y-4X=3Y-2X
HENCE
2X-3Y+Z=0 IS THE REQD. EQN.