Page 65 - 35Linear Algebra

2.5 Solution Sets for Systems of Linear Equations 65

1. Unique Solution. The planes have a unique point of intersection.

2a. No solutions. Some of the equations are contradictory, so no solutions
exist.

2bi. Line. The planes intersect in a common line; any point on that line
then gives a solution to the system of equations.

2bii. Plane. Perhaps you only had one equation to begin with, or else all
of the equations coincide geometrically. In this case, you have a plane
of solutions, with two free parameters.

Planes

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2biii. All of R . If you start with no information, then any point in R is a
solution. There are three free parameters.

In general, for systems of equations with k unknowns, there are k + 2
possible outcomes, corresponding to the possible numbers (i.e., 0, 1, 2, . . . , k)
of free parameters in the solutions set, plus the possibility of no solutions.
These types of solution sets are hyperplanes, generalizations of planes that
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behave like planes in R in many ways.

Reading homework: problem 4

Pictures and Explanation

2.5.2 Particular Solution + Homogeneous Solutions

Lets look at solution sets again, this time trying to get to their geometric
shape. In the standard approach, variables corresponding to columns that
do not contain a pivot (after going to reduced row echelon form) are free. It
is the number of free variables that determines the geometry of the solution
set.

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