Linear Combinations: Will Two Always Produce b=(0,1)?

[porschedude]

Will there always be two different combinations that produce b=(0,1) of three vectors: u, v, and w?

I'm pretty certain that the answer is no, but am I right in saying that with three vectors, assuming they are not all parallel, will always have at least one combination that produces (0,1)

 

[Dick]

You've got that right. If you have three vectors in R^2 then either there are no combinations that produce (0,1) (if they are all parallel and not parallel to (0,1)) or there are an infinite number.

 

[porschedude]

How can there be an infinite number?

 

[Dick]

How can there be an infinite number?

Say u=(0,1), v=(1,0) and w=(1,1). There are an infinite number of solutions to the equation a*u+b*v+c*w=(0,1). You should be able to show that. It's maybe a little harder to show that if there is one solution, then there are an infinite number.