Mathematics 244 – Linear Algebra Discussion 2 – More on Systems of Linear Equations February 11, 2004

Background

We have seen a systematic approach to solving systems of linear equations by applying elementary operations to put the system into echelon form. We can then parametrize the set of solutions (if there are any) by solving for the basic variables of the system (the ones appearing in leading terms in the echelon form system) in terms of the free variables. In the text, we often substitute values or other parameters for the free variables to get the parametrization. Today we want to do some more examples and think about general patterns here.

Discussion Questions

A) Put the following system into echelon form:

\begin{aligned}x_{1} & + & 3x_{2} & + & x_{3} & = & 1 \\ x_{1} & − & x_{2} & + & x_{3} & = & 2 \\ 3x_{1} & + & x_{2} & + & 3x_{3} & = & 4 \end{aligned}

What do your results mean? Explain why it would be reasonable to call this system inconsistent.

B) In this problem we will start to consider a connection between homogeneous and inhomogeneous systems of equations:

1) Find a linearly independent set S which spans the subspace W of R^{5} consisting of all solutions of the following system of equations:

2x_{1} + 3x_{2} +  + x_{4} − x_{5} = 0
x_{1} + x_{3} + x_{5} = 0
x_{1} + x_{2} + x_{3} − x_{4}  = 0

2) Now consider a system of equations with the same left hand sides, but the following nonzero right hand sides:

2x_{1} + 3x_{2} +  + x_{4} − x_{5} = 1
x_{1} + x_{3} + x_{5} = 2
x_{1} + x_{2} + x_{3} − x_{4}  = 3

Find a parametrization for the set of solutions.

3) How are your answers in parts 1 and 2 related? Try to relate this to the construction of y + W from problem 5 of Section 1.2 in the book (from Problem Set 1).

C) Decide whether each of the following statements is true or false. For the true ones, give a proof. For the false ones, give a counterexample.

1) Every homogeneous system of linear equations with more variables than equations (that is, m < n) has a solution different from the zero vector.

2) Every system of linear equations has at least one free variable.

3) If the number of equations is greater than the number of variables (m > n), then a system of linear equations has no solutions.