Linear Algebra - Historical Background, Fundamental Principles, Matrices, Applications - Vectors
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. Vectors correspond to directed line segments. They have both magnitude (length) and direction. Matrices are rectangular arrays of numbers. They are used in dealing with the coefficients of simultaneous equations. Using vector and matrix notation, a system of linear equations can be written, in the form of a single equation, as a matrix times a vector.
Linear algebra has a wide variety of applications. It is useful in solving network problems, such as calculating current flow in various branches of complicated electronic circuits, or analyzing traffic flow patterns on city streets and interstate highways. Linear algebra is also the basis of a process called linear programming, widely used in business to solve a variety of problems that often contain a very large number of variables.
Since the solution to a system of simultaneous equations, as pointed out earlier, corresponds to the point in space where their graphs intersect in a single point, and since vectors represent points in space, the solution to a set of simultaneous equations is a vector. Thus, all the variables in a system of equations can be represented by a single variable, namely a vector.
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