1 Inhomogeneous linear systems.

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1 Geometry at Lingotto. LeLing: Inhomogeneous linear systems. C ontents: Inhomogeneous linear systems. Gauß-Jordan for inhomogeneous systems. The general solution. General solution as sum of the homogeneous solution plus a particular solution. Geometric meaning of solutions. R ecommended exercises: GeoLing,5. Inhomogeneous linear systems. A system of equations of the form: S = a x + a a n x n = b a 2 x + a a 2 n x n = b 2 a x + a a n x n = b a m x + a m a m n x n = b m with at least one b j different from zero is called inhomogeneous or non-homogeneous system of linear equations in n unknonws and m equations. The zero column =. Example.. Two examples: is never a solution of an inhomogeneous system. A = { x y = x + y = B = { x + x 4 = x 5 = 2 The system is not hard to remember if one uses the augmented matrix : Ingegneria dell Autoveicolo, LeLing Geometry

2 . The notion of solution. Geometry at Lingotto. a a 2 a n a 2 a 2 2 a 2 n a a 2 a n.... a m a m 2 a m n The matrix A is the coefficient matrix: a a 2 a n a 2 a 2 2 a 2 n a a 2 a n.... a m a m 2 a m n and the last column of the augmented matrix is denoted with B = The symbol (A B is used to indicate the augmented matrix of an inhomogeneous system having A as coefficient matrix and B as the above mentioned column. b b 2 b. b m b b 2 b. b m. Example.2. The augmented matrices of the previous examples are: ( ( Note the relation between the unknown x and the first column of A, and the second column of A, etcetera, and at last, the relation between the equations of S and the rows of A. Remark.. { Be extremely careful( when writing the matrix associated to a system. x + y = The matrix of is not y + x = The notion of solution. The most striking difference between homogeneous and inhomogeneous systems concerns the existence of a solution (the zero column is not a solution in the inhomogeneous 2 Ingegneria dell Autoveicolo, LeLing 2 Geometry

3 .2 Equivalent systems and EROs. Geometry at Lingotto. case, by definition!!. That is to say, inhomogeneous system might admit no solution at all, like in this example: I = { x = x =. is: ( It is a system on equation in unknown that clearly has no solution. Its matrix Definition.4. A linear system is inconsistent if it has no solution. Otherwise the system is called consistent. Later we will explain how to decide whether a system is consistent or not. As for the homogeneous systems, a solution to a system S is a column R = (r i whose elements satisfy each equation of the system S when substituted to the unknowns x i. Solutions are thus written as columns..2 Equivalent systems and EROs. As for homogeneous systems, S and S are said equivalent if they admit the same solutions. The elementary row operations ERO, ERO2, ERO can be used to generate equivalent systems starting from a given system. Once again one works on the system s matrix. The idea is to simplify the system using EROs.. Gauß-Jordan elimination. The Gauß-Jordan method can be adapted to solve inhomogeneous systems as well. We simply act on the augmented matrix as if dealing with a homogeneous system, using the Gaußstep only on the coefficient matrix A. Ingegneria dell Autoveicolo, LeLing Geometry

4 . Gauß-Jordan elimination. Geometry at Lingotto. ( Example.5. We show how to solve the system with matrix. We find ( ( ( R 2 R = R 2 /2 2 = /2 Here ends the Gauß step and begins the Jordan step. Then ( /2 R +R 2 = ( /2 /2 At the end x = /2 and y = /2, so the solution is the column ( /2 /2. The vertical line separating the matrix coefficient from the column B tells where to end the Gauß step. It may happen to have to solve many systems simultaneously. If so, we may just juxtapose the columns B of the various systems after th vertical line and proceed as before. For instance: Example.6. The systems { x + x 4 + x 5 = x 5 = { x + x 4 + x 5 = 4 x 5 = 7 have a common coefficient matrix and can therefore be solved at he same time by considering: ( ( / / With R / we get and ends Gauß. By R R 2 we find ( / / 4 7 / 4/ 7 4/ 7/ 7 and finish the Jordan step. The equivalent systems obtained are: { x / + /x 4 + = 4/ x 5 = { x / + /x 4 + = 7/ x 5 = 7 Ingegneria dell Autoveicolo, LeLing 4 Geometry

5 .4 General solution Geometry at Lingotto. The solutionscome from assigning to, x, x 4, arbitrary values and finding x, x 5 x x 4 7 from the system x x 4 for the first system and x x for the second..4 General solution The solutions of is a sum of two { x + x 4 + x 5 = x 5 = x 4 x x 4 are + x x x 4 4. This column where the former involves the parameters, while the latter does not depend on the unknowns and is called particular solution. To a non-homogeneous system S with matrix (A B one can associate a homogeneous system: just take A as matrix of a homogeneous system. Theorem.7. Let S be an inhomogeneous system with matrix (A B. Call X an arbitrary solution of S. Then every solution of S can be expressed as: X + Y where Y is the solution of the associated homogeneous system. So the knowledge of an inhomogeneous solution allows to find all solutions by solving the associated homogeneous system. Ingegneria dell Autoveicolo, LeLing 5 Geometry

6 .4 General solution Geometry at Lingotto. Example.8. The column 2 solves the system S = The solutions of S can be written as: { x + y + z = 6 2x y z = { x + y + z = 2x y z = x y z with Ingegneria dell Autoveicolo, LeLing 6 Geometry

7 .5 Geometric interpretation. Geometry at Lingotto..5 Geometric interpretation. A linear equation ax = b represents the point x = b a of the real line, provided a. has x = as so- Example.9. The system { x = lution, a point of the real line. Similarly, a linear equation ax+by = c in two unknowns x, y represents a straight line in the plane, if a or b. Example.. The system { x + 2y = a straight line in the plane. has as solution Any equation of a system with two variables can be understood geometrically as a line in the plane. If these lines intersect at a common point the system is consistent, otherwise it will be inconsistent. { ( y x = x The solution of is y + 2x = 4 y hence the intersection point of the two lines. = ( 2, Take the inconsistent system are parallel, hence do not meet. { y x = y x =. The lines Ingegneria dell Autoveicolo, LeLing 7 Geometry

8 .5 Geometric interpretation. Geometry at Lingotto. The equation x+y +z =, with three variables, is a plane in space. The inconsistent system { z = z = has no solution: At last, here is the line solution of a consisten system, seen as intersection of two planes in space: Ingegneria dell Autoveicolo, LeLing 8 Geometry