Linear Algebra really ought to be called Linear Geometry , because it is about the geometry of the simplest straight or flat spaces.
Traditions being what they are, it is a bit late to change the name now.
To give some intuitive, geometric idea of what this means, is the number line, thought of as an infinitely long, infinitely thin ruler which somehow has numbers marked on it. The most practical reasons are very basic indeed. When a scientist or engineer measures the state of a system, he or she comes out with a list of numbers. If the system is the rather simple one of a single ball rolling over a table, at any time he measures the position of the ball on the table. He probably uses the standard Cartesian coordinates , usually denoted by x and y for historical reasons.