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Let us call a simplicial complex a complex of linear independencies if its vertices can be mapped to (non-zero elements of) a linear (vector) space so that a finite set of vertices span a simplex if and only if their images are linearly independent.

Do such simplicial complexes have an official name? Do they have special properties or interesting applications? Have they been studied?

Note that not every simplicial complex is a complex of linear independencies. For example, a simplicial complex with three $0$-dimensional simplices (vertices) and a single $1$-dimensional simplex (edge) is not.

P.S. I suppose one can also define and study simplicial complexes of algebraic independencies, etc.

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    I'm outside my area of expertise here, but I think this might be connected to the idea of a Matroid: https://en.wikipedia.org/wiki/Matroid – 2017-01-25
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    It seems like this might be the answer. – 2017-01-25
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    Oscar is correct I think, since sets of linearly independent vectors satisfy the exchange property and are independence systems (also called abstract simplicial complexes), which is what the definition of a matroid is (I think). – 2017-02-17
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    @Alexey Thanks! – 2017-02-17

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These are examples of Matroids (although not every matroid arises in this way).