The collection of theorems and ideas that comprise linear algebra have come together over some four centuries, beginning in the mid 1600s. The name linear algebra, however, is relatively recent. It derives from the fact that the graph of a linear equation is a straight line. In fact the beginnings of linear algebra are rooted in the early attempts of sixteenth and seventeenth century mathematicians to develop generalized methods for solving systems of linear equations. As early as 1693, Gottfried Leibniz put forth the notion of matrices and their determinants, and in 1750, Gabriel Cramer published his rule (it bears his name today) for solving n equations in n unknowns.
The concept of a vector, however, was originally introduced in physics applications to describe quantities having both magnitude and direction, such as force and velocity. Later, the concept was blended with many of the other notions of linear algebra when mathematicians realized that vectors and one column (or one row) matrices are mathematically identical.
Finally, the theory of vector spaces grew out of work on the algebra of vectors.