The theory of error correcting codes is a very nice and elementary context for introducing linear algebra, assuming the students know $\mathbb{F}_2$. The notion that each message of bit-length $n$ can coded as a "vector" over $\mathbb{F}_2$ of dimension $m > n$, using some linear conditions ("linear subspace") so as to provide easy error-checking conditions, should be quite motivating. Concepts such as "linear transforms" (matrices) and "null-spaces" show up naturally when considering the parity check matrix of the code. Etc., Etc.
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