Branch of algebra concerned with methods of solving systems of linear linear transformations and vector spaces .
"Linear" refers to the form of the equations involved
in two dimensions, a x + b y = c . Geometrically, this represents a line. If the variables are replaced by vectors , functions , or derivatives , the equation becomes a linear transformation. A system of equations of this type is a system of linear transformations. Because it shows when such a system has a solution and how to find it, linear algebra is essential to the theory of mathematical analysis and {{link=differential equation">differential equations . Its applications extend beyond the physical sciences into, for example, biology and economics.