Linear algebra belongs to standard undergraduate curricula in studies of mathematics, sciences and engineering and our paper aims to contribute to the research in teaching undergraduate linear algebra with two parts. First, we describe a non-standard approach to teaching vector algebra and analytic geometry in skew coordinate systems. We describe how the crystallographic reciprocal space construct leads to a simple generalization of calculations of normal vectors to planes in general skew coordinate systems, and also leads, purely mathematically, to the crystallographic fundamental law of the reciprocal lattice. Second, insights obtained from more than ten years long experience of teaching courses that include linear algebra topics to chemistry and geology students are stated, and related to current research in teaching linear algebra. Finally, conjectures for future research are formulated.