Linear Equations in Linear Algebra

Transcription

1 1 Linear Equations in Linear Algebra 1.5 SOLUTION SETS OF LINEAR SYSTEMS

2 HOMOGENEOUS LINEAR SYSTEMS A system of linear equations is said to be homogeneous if it can be written in the form A x= 0, where A is an m nmatrix and 0 is the zero vector in m. Such a system A x= 0always has at least one solution, namely, x= 0 (the zero vector in n ). This zero solution is usually called the trivial solution. The homogeneous equation A x= 0has a nontrivial solution if and only if the equation has at least one free variable. Slide 1.5-2

3 HOMOGENEOUS LINEAR SYSTEMS Example 1: Determine if the following homogeneous system has a nontrivial solution. Then describe the solution set. 3x + 5x 4x = x 2x + 4x = x + x 8x = Solution: Let A be the matrix of coefficients of the system and row reduce the augmented matrix A 0 to echelon form: [ ] Slide 1.5-3

4 " # HOMOGENEOUS LINEAR SYSTEMS % " & # A x = 0 % " & # Since x 3 is a free variable, has nontrivial solutions (one for each choice of x 3.) Continue the row reduction of echelon form: [ A 0] x to reduced 4 x = 0 3 x = = 0 % & Slide 1.5-4

5 HOMOGENEOUS LINEAR SYSTEMS 4 Solve for the basic variables x 1 and x 2 to obtain x = x 1 3, 3 x = 0, with x 3 free. 2 As a vector, the general solution of A x = 0 has the form given below. 4 4 x3 x1 3 3 x = x = 0 = x 0 = x v x x 1 3 3, where 4 3 v= 0 1 Slide 1.5-5

6 HOMOGENEOUS LINEAR SYSTEMS Here x 3 is factored out of the expression for the general solution vector. This shows that every solution of a scalar multiple of v. in this case is The trivial solution is obtained by choosing. Geometrically, the solution set is a line through 0 in. See the figure below. A x = 0 x = Slide 1.5-6

7 PARAMETRIC VECTOR FORM x= su+ tv The equation of the form (s, t in ) is called a parametric vector equation of the plane. In Example 1, the equation (with x 3 free), or 3 x= tv(with t in ), is a parametric vector equation of a line. Whenever a solution set is described explicitly with vectors as in Example 1, we say that the solution is in parametric vector form. x = x v Slide 1.5-7

8 When a nonhomogeneous linear system has many solutions, the general solution can be written in parametric vector form as one vector plus an arbitrary linear combination of vectors that satisfy the corresponding homogeneous system. A x= 7 b= 1 4 Example 2: Describe all solutions of A = and. b, where Slide 1.5-8

9 Solution: Row operations on " # x " % & # 4 = 1+ x x = [ A 0] produce Thus,, and x 3 is free. 1 3 % & x 1 4, 3 x = 1 3. x 2 = 2 0 = 0 Slide 1.5-9

10 As a vector, the general solution of given below. A x= b has the form x x 3 3 x x = x = 2 = 2 = 0 = 2 + x x x 0 x p v Slide

11 The equation parameter, x= p+x v, or, writing t as a general describes the solution set of vector form. The solution set of equation x= p+ tv x (t in ) ----(1) in parametric has the parametric vector (t in ) ----(2) [with the same v that appears in (1)]. 3 A x= 0 = tv A x= b A x= b Thus the solutions of are obtained by adding the vector p to the solutions of A x= 0. Slide

12 The vector p itself is just one particular solution of [corresponding to t = in (1).] A x= b 0 Now, to describe the solution of A x= bgeometrically, we can think of vector addition as a translation. 2 3 Given v and p in or, the effect of adding p to v is to move v in a direction parallel to the line through p and 0. Slide

13 We say that v is translated by p to v+ p. See the following figure. 2 3 If each point on a line L in or is translated by a vector p, the result is a line parallel to L. See the following figure. Slide

14 Suppose L is the line through 0 and v, described by equation (2). Adding p to each point on L produces the translated line described by equation (1). We call (1) the equation of the line through p parallel to v. Thus the solution set of A x= bis a line through p parallel to the solution set of A x= 0. The figure on the next slide illustrates this case. Slide

15 A x= b A x= 0 The relation between the solution sets of and shown in the figure above generalizes to any consistent equation A x= b, although the solution set will be larger than a line when there are several free variables. Slide

16 Theorem 6: Suppose the equation A x= b is consistent for some given b, and let p be a solution. Then the solution set of A x= bis the set of all vectors of the form w= p+ v h, where v h is any solution of the homogeneous equation A x= 0. This theorem says that if A x= bhas a solution, then the solution set is obtained by translating the solution set of A x= 0, using any particular solution p of A x= bfor the translation. Slide

17 WRITING A SOLUTION SET (OF A CONSISTENT SYSTEM) IN PARAMETRIC VECTOR FORM 1. Row reduce the augmented matrix to reduced echelon form. 2. Express each basic variable in terms of any free variables appearing in an equation. 3. Write a typical solution x as a vector whose entries depend on the free variables, if any. 4. Decompose x into a linear combination of vectors (with numeric entries) using the free variables as parameters. Slide