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Unformatted text preview: Unit 17 The Theory of Linear Systems In this section, we look at characteristics of systems of linear equations and also of their solution sets. Theorem 17.1. For any system of linear equations Avectorx = vector b , there are only 3 possibilities: the system has no solution, or the system has exactly one solution, or the system has infinitely many solutions (i.e. a parametric family of solutions). Proof: We have seen examples of each kind. We need to show that there are no other possibilities. That is, we need to show that any system which has more than one solution must have infinitely many solutions. Consider any SLE Avectorx = vector b and suppose that there exist 2 distinct solutions to the system, vector y and vector z . That is, we have Avector y = vector b and Avector z = vector b with vector z negationslash = vector y . Now, consider a vector vectorw = (1- t ) vector y + t ( vector z ) for any t Rfractur . Then Avectorw = A [(1- t ) vector y + tvector z ] = A (1- t ) vector y + Atvector z = Avector y- Atvector y + Atvector z = (1- t )( Avector y ) + t ( Avector z ) (By Property (n) of matrix operations, text p. 102) = (1- t ) vector b + t vector b (since Avector y = vector b and also Avector z = vector b ) = vector b- t vector b + t vector b = vector b We see that Avectorw = vector b as well. That is, if there exist 2 distinct solutions, vector y and vector z , to Avectorx = vector b , then any vector of the form vectorw = (1- t ) vector y + tvector z for t Rfractur is also 1 a solution to Avectorx = vector b , so there are, in fact, infinitely many solutions. Thus we see that any SLE which has more than one solution always has infinitely many solutions. Definition 17.2. A SLE of the form Avectorx = vector 0 is called a homogeneous system. A system of the form Avectorx = vector b with vector b negationslash = vector 0 is called non-homogeneous . Example 1 . Identify the following systems as homogeneous or non-homogeneous. (a) 1- 1 2 0 1 2 0 6 3 2 7 x 1 x 2 x 3 x 4 = homogeneous system. (b) 3 x 1 + 2 x 2 + x 3 = 3 x 2- x 3 = homogeneous system. 2 x 1 + x 3 = (c) x 1- 2 x 2 + 3 x 3 = x 1- 2 x 3 = non-homogeneous system. 5 x 2 + x 3 = 1 Notice: The zero vector, vector 0, is a solution to every homogeneous system Avectorx = vector 0, since A vector 0 = vector 0 for any coefficient matrix A . (Note that in the statement A vector 0 = vector 0, the zero-vectors are not necessarily the same size. If A has dimension m n , then the vector 0 on the left side of the equation is an n-vector, while the vector 0 on the right side is an m-vector. That is, any homogeneous system Avectorx = vector 0 has vector Rfractur n as a solution, where, of course, n is the number of variables in the system.) Definition 17.3. For any homogeneous system Avectorx = vector 0, the solution vectorx = vector is referred to as the trivial solution . Any other solutions are called non-trivial...
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