The study of Linear Algebra includes the topics of vector algebra, matrix algebra, and the theory of vector spaces. Linear Algebra originated as the study of linear equations, including the solution of simultaneous linear equations. An equation is linear if no variable in it is multiplied by itself or any other variable. Thus, the equation 3x + 2y + z = 0 is a linear equation in three variables. The equation x³ + 6y + z + 5 = 0 is not linear, because the variable x is raised to the power 3 (multiplied together three times); it is a cubic equation. The equation 5x - xy + 6z = 7 is not a linear equation either, because the product of two variables (xy) appears in it. Thus linear equations are always degree 1.

Two important concepts emerge in Linear Algebra to help facilitate the expression and solution of systems of simultaneous linear equations. They are the vector and the matrix...