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Linear algebra is a branch of mathematics. It came from mathematicians trying to solve systems of linear equations. Vectors and matrices are used to solve these systems. The main objects of study currently are vector spaces and linear mappings between vector spaces. Linear algebra is useful in other branches of mathematics (e.g. differential equations and analytic geometry). It can also be applied to the real world in areas such as engineering, physics and economics.
For example consider the following equations:
These two equations form a system of linear equations. It is linear because none of the variables are raised to a power. The graph of a linear equation in two variables is a straight line. The solution to this system is: