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Unformatted text preview: Lecture 15 Andrei Antonenko March 07, 2003 1 Homogeneous systems Last time we studied linear functions. To continue our theory about them we have to study another topic, which we will use in our future lectures. Recall, that the system is called homogeneous, if it has 0s in the right hand side. The main fact about them is that each homogeneous system has either 1 or infinitely many solutions; moreover, if the number of equations is less than the number of variables, then the system has infinitely many solutions. Well state some properties of the solution set of the homogeneous system. Existence of 0. (0 , ,..., 0) is a solution. Addition of solutions. Let we have 2 solutions of the homogeneous system, ( y 1 ,y 2 ,...,y n ) and ( y 1 ,y 2 ,...,y n ). Then ( y 1 + y 1 ,y 2 + y 2 ,...,y n + y n ) is a solution. To check it consider an equation of the system: a i 1 x 1 + a i 2 x 2 + + a in x n = 0 . Since specified n-tuples are solutions, we have: a i 1 y 1 + a i 2 y 2 + + a in y n = 0 and a i 1 y 1 + a i 2 y 2 + + a in y n = 0 . Adding these 2 equations, we have: a i 1 ( y 1 + y 1 ) + a i 2 ( y 2 + y 2 ) + + a in ( y n + y n ) = 0 , so, ( y 1 + y 1 ,y 2 + y 2 ,...,y n + y n ) is a solution. Multiplication by a number. Let ( y 1 ,y 2 ,...,y n ) be a solution. Then ( cy 1 ,cy 2 ,...,cy n ) is a solution. To check it consider an equation of the system: a i 1 x 1 + a i 2 x 2 + + a in x n = 0 ....
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This note was uploaded on 05/28/2011 for the course AMS 2010 taught by Professor Andant during the Spring '03 term at SUNY Stony Brook.
- Spring '03