School

University of Waterloo
Department

Mathematics

Course Code

MATH136

Professor

Patrick Roh

Lecture

7

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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