Elementary applications of linear algebra over finite fields

How about binary linear codes? You can "see" the Hamming distance between codewords, and use linear transformations to encode/decode

You can use linear algebra over $\mathbb{F}_2$ to solve the game "Lights Out": http://en.wikipedia.org/wiki/Lights_Out_%28game%29

I suggest Linear Feedback Shift Register (LFSR) as an easy example. They can be used as pseudo random number generators and have a wide practical use in communication and cryptography, GPS, GSM, CRC, WIFI, .. (non-math) applications which are usually accepted as usefull.

Usually they work over $\mathbb{F}_2$, but other fields are possible. Basically you have to work with polynomials (including long division) over $\mathbb{F}_2$. The need for primitive polynomials may motivate some more advanced considerations. A brief summary for mathematicians is Nayuki's blog.

I would explicitly pick the CRC algorithm. A description is located for example in this lecture(pdf) from D.Culler. This also relates to linear codes, which is also a good idea.

More easy is an application as fancy counter. If you ever wondered how the shuffle mode of your media player works.