solution vectors for sparse linear equation systems

The non-zero elements are shown in black. The equations of a linear system are independent if none of the equations can be derived algebraically from the others.A sparse matrix obtained when solving a finite element problem in two dimensions. It is possible for a system of two equations and two unknowns to have no solution (if the two lines are parallel), or for a system of three equations and two unknowns to be solvable (if the three lines intersect at a single point).Ī system of linear equations behave differently from the general case if the equations are linearly dependent, or if it is inconsistent and has no more equations than unknowns. It must be kept in mind that the pictures above show only the most common case (the general case). The third system has no solutions, since the three lines share no common point. The second system has a single unique solution, namely the intersection of the two lines. The first system has infinitely many solutions, namely all of the points on the blue line. The following pictures illustrate this trichotomy in the case of two variables: In the first case, the dimension of the solution set is, in general, equal to n − m, where n is the number of variables and m is the number of equations. Such a system is also known as an overdetermined system.

In general, a system with more equations than unknowns has no solution.

In general, a system with the same number of equations and unknowns has a single unique solution.

Such a system is known as an underdetermined system.

In general, a system with fewer equations than unknowns has infinitely many solutions, but it may have no solution.Here, 'in general' means that a different behavior may occur for specific values of the coefficients of the equations. In general, the behavior of a linear system is determined by the relationship between the number of equations and the number of unknowns. The solution set for two equations in three variables is, in general, a line. The solution set is the intersection of these hyperplanes, and is a flat, which may have any dimension lower than n. įor n variables, each linear equation determines a hyperplane in n-dimensional space. For example, as three parallel planes do not have a common point, the solution set of their equations is empty the solution set of the equations of three planes intersecting at a point is single point if three planes pass through two points, their equations have at least two common solutions in fact the solution set is infinite and consists in all the line passing through these points. Thus the solution set may be a plane, a line, a single point, or the empty set. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set.įor three variables, each linear equation determines a plane in three-dimensional space, and the solution set is the intersection of these planes. The system has a single unique solution.įor a system involving two variables ( x and y), each linear equation determines a line on the xy- plane.The system has infinitely many solutions.Ī linear system may behave in any one of three possible ways: The set of all possible solutions is called the solution set. , x n such that each of the equations is satisfied. The solution set for the equations x − y = −1 and 3 x + y = 9 is the single point (2, 3).Ī solution of a linear system is an assignment of values to the variables x 1, x 2. The number of vectors in a basis for the span is now expressed as the rank of the matrix.

A linear system in three variables determines a collection of planes The intersection point is the solution.

Solution vectors for sparse linear equation systems