MAT 2401 Linear Algebra 1.2 Part II Gauss-Jordan Elimination

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Presentation transcript:

MAT 2401 Linear Algebra 1.2 Part II Gauss-Jordan Elimination

HW Written Homework

Time Part I may be a bit longer. Part II will be shorter.

Preview System with No solutions. System with Infinite Number of solutions.

How many solutions? Q: Given a system of 3 equations in 3 unknowns, how many solutions are possible? Inconsistent System Consistent System

Example 4 Use Gauss-Jordan Elimination to solve the system. Conclusion: 123

Example 5 Use Gauss-Jordan Elimination to solve the system.

Example 5 123

Geometric Interpretation We are looking for the intersection points of 3 planes (linear equations) It turns out that they are the same intersection points of (another) 2 planes  A straight line

Remarks The solution set is the collection of the points on the intersection line of the planes: x-3z=-1 and y-z=0.

Expectations Some descriptions are necessary to help your audience to follow your solutions Here, I suggested “The system becomes”. You can use similar wordings if you want.

Expectations Since the intersection is a line, it is customary to represent it as a parametric equations (Calculus III)

Expectations Instead of “t is any real number”, we use the set notation i.e. t is an element of the real numbers.

System of Linear Equations (LE)

Homogeneous System of LE

Q: Is the system consistent?A:

Trivial and Non-Trivial Solutions Trivial Solutions Non-Trivial Solutions At least one of the x i is non-zero. (You need this for today’s HW)

Visual Summary