Solution Sets of Linear Systems (9/21/05)

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Solution Sets of Linear Systems (9/21/05)
A system of linear equations is called homogeneous if it can be written in the form of the matrix equation A x = 0, i.e., all the constants are 0. Note that such a system can never be inconsistent (why??). Hence there is always at least one solution, namely the 0 vector. If 0 is the unique solution, then there are no free variables.

Homogeneous Systems If a homogeneous system has exactly one free variable xn (say), then the solution set can be written in parametric vector form xn v (or more commonly t v) for a single vector v. Geometrically, this solution set is a line passing through the origin. Do an example….

More Homogeneous Systems
If a homogeneous system has two free variables, then the solution set will be a linear combination of two vectors, i.e., it will be t v + s w where t and s are parameters (each ranging over all reals). Geometrically, this solution is…… Again look at an example…

Nonhomogeneous Systems
Such a system A x = b (where b is not 0) may (of course) be inconsistent. If such a system is consistent and if the vector p is any single solution, then the general solution is just p + the solution set of the corresponding homogeneous system A x = 0. Geometrically, this is translation by p.

Writing a Solution in Parametric Vector Form
Row reduce the augmented matrix as usual. Express all variables in terms of the free variables (if any). Change this expression into parametric vector form, i.e., with the column of all variables on the left and with columns of numbers with the free variables as parameters on the right.

Applications, and the Assignment for Friday
Section 1.6 covers three applications of linear systems: input-output models in economics, balancing equations in chemistry, and network flow theory. You’re welcome to read it all, but do read the section on balancing chemical equations. For Friday, please read Section 1.5 and do the Practice and Exercises 1, 3, 5, 7, 13, 15, 19, 21, 23. Also do Exercise 7 on page 63 (in Section 1.6).