if one linear combination of 3 vectors produces b, then infinitely many combinations will produce b The question is: If I take any three vectors u, v, w in the xy plane, will there always be different combinations that produce b=(0,1)?
The solution (also includes the answer to another part of the question):

I'm having difficulty understanding how "if one combination produces b then two (and infinitely many) combinations will produce b." How are there infinitely many ways? I can see one where the 3rd is the zero vector, but not infinite. 
 A: No, because $u=(1,0); v=(2,0); w=(3,0)$ won't work.
But if the vectors span the plane, two of them form a basis, say $u,v$ (call the third $w\neq 0$) and there will be real numbers $a,b$ with $au+bv=(0,1)$ and you will also have (non-zero) $c,d$ with $cu+dv-w=(0,0)$. Any multiple of the zero vector is zero.
In the case $w=0$ there is also an easy answer.
A: If $u, v$ and $w$ are three non-zero vectors in the x-y plane, then they are linearly dependent.  In other words, one of them (say $u$) can be expressed as a linear combination of the other two ($v$ and $w$).   This is true because of a fact from elementary linear algebra that says the maximum number of linearly independent vectors in $n$-dimensional space is $n$.  For the x-y plane, $n=2$.  
Suppose that some linear combination of $u, v$ and $w$ is equal to $b=(0,1)$: $b=c_1 u+ c_2 v+c_3 w$. We want to see why there exist infinitely many linear combinations of $u, v$ and $w$ that will produce the same $b$.  By the previous paragraph, $u$ can be expressed as a linear combination of $v$ and $w$, $u=d_1 v+ d_2 w$ say. Hence, we can transfer some of the contribution $c_1 u$ of $u$ to $b$ to the vectors $v$ and $w$. For example, the term $c_1 u$ can be reduced to $(c_1 - \epsilon) u$, while the remaining $\epsilon u$ which needs to be added back can be expressed in terms of $v$ and $w$.  In this manner, by choosing different values for $\epsilon$, we get different linear combinations.
A: We have the freedom to scale the vectors by chosen constants, and perhaps this example will demonstrate how we can find an infinite set of constants, which gives arise to infinitely many linear combinations.
Consider $\mathbf{u} = (1,0), \mathbf{v} = (0,1)$, and $\mathbf{w} = (1,1)$. Then $\mathbf{b} = c_1\mathbf{u} + c_2\mathbf{v} + c_3\mathbf{w}  = (0,1)$ for a chosen set $\{c_1,c_2,c_3\}$. The trivial choice is $\{c_1,c_2,c_3\} = \{0,1,0\}$ but we can also choose $\{c_1,c_2,c_3\}$ to be
$\quad\{1,2,-1\}$
$\quad\{2,3,-2\}$
$\quad\{3,4,-3\}$
and so on...
$\quad\{k,k+1,-k\}$
