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1. the vectors u and v span R^2.
a,b,c,d E R.
show that au+bv and cu+dv span R^2 iff ad-bc does not equal 0.
2. (Original post by realicetic)
the vectors u and v span R^2.
a,b,c,d E R.
show that au+bv and cu+dv span R^2 iff ad-bc does not equal 0.
au+bv, cu+dv span R^2 iff they are linearly independent.
i.e. s(au+bv)+t(cu+dv)=0 => s=t=0.
i.e. u(as+ct)+v(bs+dt)=0 => s=t=0
Since u and v are linearly independent, it follows that s=t=0 must be the only solution to:
as+ct=0
bs+dt=0
Write this equation in the form Ax=0, then use the fact that this equation has only the zero solution iff A is invertible.

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Updated: February 27, 2006
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