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Lecture 7

Week 3 (lecture 7-9) Lecture Notes

by OneClass31850 , Winter 2012
10 Pages
135 Views

Department
Mathematics
Course Code
MATH136
Professor
Patrick Roh
Lecture
7

This preview shows pages 1-3. Sign up to view the full 10 pages of the document.
Eg
Find the shortest distance from the point to the plane 
Note:
This plane does NOT pass through the origin.
The vector

does NOT go from the plane to .
The normal vector of the plane is

We need a vector from ANY point on the plane to .
The point is on the plane











Systems of linear equations
Definition
An equation of the form for constants is called a linear equation.
The constants are the coefficients of the variables .
A set of linear equations is called a system of linear equations.
(Note: A linear equation can be represented geometrically as a hyperplane.)
n
Q


Lec 7 - Shortest paths and System of Linear Equations
Wednesday, January 18, 2012
9:33 AM
MATH 136 Page 1
Eg
is a linear equation.
is NOT a linear equation.
is a linear equation. (should be written as .)
(This is known as the standard form)
In general, we will be looking at systems of linear equations in variables:



Where is the coefficient of variable in the equation .
Any vector
is a solution of the system if the equations are satisfied when
Definition
A system of linear equations is consistent if there is at least one solution.
Otherwise it is inconsistent.
Eg
This system has
as a solution.
The system is consistent.
Geometrically, we have two planes in that intersect. The point is one of the points of intersection.
Eg


No solutions. System is inconsistent.
Geometrically, we have two lines in that are parallel and thus do NOT intersect.
In general, a solution of a system of linear equations in variables is a point in where hyperplanes represented
by the linear equations, intersect.
We could have a system with
No solution. (Inconsistent)
1.
Eg parallel hyper planes.
One solution.
2.
All the hyperplanes intersect at a single point.
Eg Intersection of the planes is a line.
More than one solution
3.
Can you create a system with exactly 2 solutions?
MATH 136 Page 2
Can you create a system with exactly 2 solutions?
Theorem: If the following system of linear equations:



has two distinct solutions
and
Then

is a solution for any and each gives a different solution.
(A consistent system with more than one solution has infinitely many solutions.)
Proof:
For each equation :










is a solution to the system for any
.
Assume , but
Thus


Since
then , a contraction.
Thus each gives a different solution.
MATH 136 Page 3

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Description
Lec 7 Shortest paths and System of Linear EquationsWednesday January 18 2012933 AMEgFind the shortest distance from the point to the plane NoteThis plane does NOT pass through the origin The vector does NOT go from the plane to Qn The normal vector of the plane is We need a vector from ANY point on the plane toThe pointis on the planeSystems of linear equationsDefinitionAn equation of the formfor constantsis called a linear equationThe constantsare the coefficientsof the variablesA set of linear equations is called a system of linear equationsNote A linear equation can be represented geometrically as a hyperplane MATH 136 Page 1
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