**Terminology:**- homogeneous linear system, trivial and non-trivial solutions, parametric vector form, linear independence and dependence.
**Objectives:**- learn about homogeneous linear systems; learn how to write linear system solutions in parametric vector form; understand the connection between linear independence and solutions to ; learn how to test for linear independence.
**Reading Assignment:**- Chapter 1, Sections 1.5-1.6 (pages 48-64).
**Lesson Outline****Key Ideas and Discussion:**-
A homogeneous linear system always has a trivial solution
. has nontrivial solutions if there is at least one
free variable. If is the coefficient matrix after a
reduction.
of the system to reduced echelon form, the non-trivial solutions
to are linear combinations of vectors obtained from those columns
of associated with the free variables. The required vectors are
*not*quite the same as the associated columns of ; 1's or 0's need to be inserted in the places for the free variables. An easy way to determine the parametric form is to write out the equations for , move all of the free variable terms to the right-hand side, insert an equation of the form for each free variable, and then rewrite your solution in vector form with on the left and a vector term for each free variable (with the free variable as a multiplier) on the right. The parametric vector form for solutions to is found by adding a (particular) solution to the parametric solution (with free variables as parameters) for . An easy way to find a particular solution to is to set any free variables (in the reduced system) equal to 0, and then determine the basic variable values from the right-hand sides.Linear independence is sometimes difficult to understand initially, and you might find it easier to think first about dependence. A set of vectors is a dependent set if one of the vectors can be written as a linear combination of ( depends on) the other vectors in the set. The simplest check for dependence is to see if any vector in the set is a multiple of one of the other vectors; if so, the set is dependent. If not, a more careful analysis is needed to decide whether the set is a dependent set or an independent set. If the vectors are put (as columns) into a matrix

*A*, then the set is dependent if and only iff has some nontrivial solutions. A general purpose check for dependence is to put the vectors into a matrix and reduce the matrix. If every column has a pivot, then the vectors are independent; otherwise they are dependent. Another simple check for dependence is to count components: if the vectors are*n*-vectors and there are more than*n*of them, they must be dependent. Unfortunately, the converse is not true: a set of*m**n*-vectors with could be dependent or independent. The columns of the identity matrix form a nice set of independent vectors. **Practice Problems:**- 1.5.3, 5, 7, 13, 21, 33 (pp. 55-57);
1.6.3, 7,
11,
17, 19, 27 (pp. 64-66).
**Assignment**-

Fri Jul 24 14:14:34 PDT 1998