MAT1360 Classwork
MAT 2401 Discovery Lab 1.2 Part II Names__________________________________
Objectives
• To explore systems of linear equations with
o no solutions
o infinite number of solutions
• To investigate the results of Gauss-Jordan Elimination in these cases.
Instructions
• Do not look up any references including the textbook and internet.
• Use correct notations and do not skip steps.
• Two persons per group. Do not communicate with other groups.
• You do not have to turn in page 1 and 5. Just make sure your names is somewhere visible.
Recall
From the last class meeting, a system of linear equations can have
(i) unique solution,
(ii) no solutions, and
(iii) infinite number of solutions.
|(i) Unique Solution |(ii) No Solutions |(iii) Infinite Number of Solutions |
|[pic] | | |
| |[pic] |[pic] |
Fig.1 Geometric interpretations of the solutions of a system of three linear equations. A solution of the system lies in the intersection of the three planes. (i) one point, (ii) no intersection, and (iii) a line.
We have looked at case (i) in details with GJE. In this lab, we are going to investigate the other two cases.
1. Solve [pic] by Gauss-Jordan Elimination. Fill in the blanks.
| |
|[pic] |
| |
|(Note that the GJE must stop here since no further steps can be carried out according to the GJE algorithm.) |
| |
|The system becomes |
|[pic] |
| |
|Conclusion: Since [pic] is impossible, the system has no solution. |
F.A.Q.
Q: I can see a “conflict” in the second step (the last two rows of the matrix). Do I have to continue with the rest of the GJE?
A: You are expected to continue the GJE until the algorithm stops.
2. Solve [pic] by Gauss-Jordan Elimination. Fill in the blanks.
| |
|[pic] |
|(Note that again the GJE algorithm cannot proceed further.) |
| |
|The system becomes [pic] or [pic]. |
| |
|Conclusion: Given each value of z, we can find the corresponding values for x and y. Thus, the system has infinity number of solutions. |
| |
|Geometrically, the solution is a line in the three-dimensional space (see the third diagram on p.1). It is customary to represent a line by a parametric equation.|
| |
|Let [pic]. Then, |
|[pic] |
| |
|(Here, t is a parameter. Also, the notation [pic] means “t is a real number”.) |
Practice
3. Solve [pic] by Gauss-Jordan Elimination.
| |
|[pic] |
|(Note that you may not need all of the matrices above. |
| |
|The system becomes [pic] or [pic]. |
| |
| |
|Let [pic]. Then, |
|[pic] |
Homogeneous System of Linear Equations
| | |
|[pic] |[pic] |
The system on the left is the general form of a system of linear equations. The system on the right is a special case where the constant terms are all zeros. It is called a Homogeneous System.
A Homogeneous System is always consistent because it has the Trivial Solution:
[pic]
(To contrast, a Non-Trivial Solution has at least one of the [pic] is non-zero.)
Visual Summary
| |
|[pic] |
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.