Quasilinear Utility Functions We know if the utility function is quasilinear (QL) w.r.t good 1, then the demand for other goods is independent of income (no income effect for goods $(2,\dots, N)$). 

But is the reverse implication also true: i.e can we say if all the goods except one have demand functions independent of income then the utility function must be quasilinear? 

I have been looking up all the standard micro grad level text books but haven't got an answer yet. I see books defining (rather characterizing) QL by just one side of implication (i.e QL implies No income effect) but are silent about the other.

Any reference in this regard will be highly useful.
Is that first claim right? I guess it depends what you mean by an income effect. 

Say your quasi-linear utility form is:
$$U(x,v) = x + \ln(v)$$
Say $P_v = P_x=1$ and so for wealth $w = P_v\cdot v + P_x \cdot x = x + v$. 
If $0&lt;w\leq 1$ the household strictly prefers to buy $v$ to $x$ and buys no $x$ but $w&gt;1$ is then exclusively $x$ is purchased beyond that. So demand for $x$ looks like this:
$$w&lt;1: x^* = 0$$
$$w&lt;1: v^* = w$$
$$w\geq1: x^* = w-1$$
$$w\geq1: v^* = 1$$

So the marginal demand for $x/v$ depends on $w$.