Linear Algebra: Real World Uses

Engineering:

Linear algebra is the simplest way to look at functions of many variables, which usually arise in engineering by the discretization of a concept stated in terms of a continuum, e.g. the law governing the relation between stresses and strains in a structure.

Linear Algebra is used quite heavily in Structural Engineering. This is for a very simple reason. The analysis of a structure in equilibrium involves writing down many equations in many unknowns. Often these equations are linear, even when material deformation (i.e. bending) is considered. This is exactly the sort of situation for which linear algebra is the best technique.
linear algebra and structural engineering

Going to Mars:

Most missions to mars are based on a Hohmann trajectory. This trajectory is the fastest trajectory possible that is minimum energy (based on the two body problem). Less energetic trajectories are available using the three body patched trajectories (Interplanetary super highway) but these can take a considerably longer length of time.

For basic mission planning the delta V of the two body problem is used to calculate the mass fraction for the trip between Earth and Mars. The mass fraction tells mission planers how much mass they need to deliver to LEO in order to get a given mass to Mars. For calculating other trajectories Numeric or algebraic techniques can be used.

Linear algebra is useful in finding the orbial plane, find out why:
linear algebra and orbital planes

Linear Programming:

Linear programming is a mathematical technique used in economics. It finds the maximum or minimum of linear functions in many variables subject to constraints.

For a more indepth definition and explanation of linear programming:
linear programming defined

A use of linear programming in economics:
economics application- profit maximization