Is every linear language in NL? I wonder if all the linear languages are in NL? 

I was thinking that we can take an input-language $L$ and convert it to linear normal form. If this is not possible, the machine rejects. If the language can be written in linear normal form, the machine accepts. The work tape only needs to store the current production, so that it works in logarithmic time.

Am I completely lost or is this a good start?
The following survey states that the result is proved by Ibarra, Jiang, Ravikumar in Information Processing Letters (1988).

Unambiguous Auxiliary Pushdown Automata and Semi-unbounded Fan-in Circuits. Niedermeier R., Rossmanith, P.,  Information and Computation 118(2), 227-245, 1995.