MAT1360 Classwork - Seattle Pacific University



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]. |

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|Conclusion: Given each value of z, we can find the corresponding values for x and y. Thus, the system has infinity number of solutions. |

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|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.|

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|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. |

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|The system becomes [pic] or [pic]. |

| |

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|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

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|[pic] |

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