9.5 - Systems of Linear Equations in More Than Two Variables A.K.A. Matrices

A Matrix is a seires of numbers within a set of brackets. These numbers are the coefficients of of each different variable called the Coefficient Matrix, and the last row is usually what the equation equals, called the Augmented Coefficient Matrix or Augmented Matrix. The last row of an augmented matrix is seprated by a line.

 Coefficient Matrix

Augmented Matrix

Definition of a Matrix

A matrix's size is determined by m and n. m is the number of rows the matrix has, and n is the number of comlumns a matrix has. 

Theorem on Matrix Row Transformations

Given a matrix of a system of linear equations, a matrix of an equivalent system results if:

1.) two rows are interchanged

2.) a row is multiplied or divided by a nonzero constant

3.) a constant multiple of one row is added to another row

Echelon Form of a Matrix

1.) The first nonzero number in each row, reading from left to right is 1.

2.) The column containing the first nonzero in any row is to the left of the column containing the first nonzero number in the row below.

3.) Rows consisting entirely of zeros may appear at the bottom of the matrix.

Guidlines for Finding the Echelon Form of a Matrix

1.) Locate the FIRST column that contains nonzero elements, and apply simple row transformations to get the number 1 into the first row of that column

2.) Apply simple row transormations of the type kR1+Rj-->Rj for j>1 to get 0 underneath the number 1 obtained in guidline 1 ineach of the remaining rows.

3.) Now, DISREGARD THE FIRST ROW. Locate the next column that contains nonzero elements, and apply simple row transformations to get the number 1 into the SECOND row of that column.

4.) Apply simple row transformations of the type kR1+Rj-->Rj for j>2 to get 0 underneath the number 1 obtained in guidline 3 in each of the remaining rows.

5.) Now, DISREGARD THE SECOND ROW. Locate the ext column that contains nonzero elements, and repeat. 

6.) Continue the process until the echelon form is reached.

Reduced Echelon Form

Reduced Echelon Form is the same as Echelon form, except in the non augmented part of the matrix there will only be 1's and 0's.

Matrices are AWESOME!!!

Austin

Chapter 6.1 ANGLES

FISRT!
Angles have two sides: the Initial Side and the Terminal Side. 

The rotation of the rays always starts on the intial side, and always ends on the terminal side.
Many angles can have the same initial and terminal sides. These are called Coterminal Angles.

When it is rotated counter-clockwise to the terminal side, the angle is positive. When it is rotated clockwise, the angle is negative.


Standard Position:
The vertex is on the origin of the graph, and the initial side is on the positive x-axis.

Finding Cotermianl Angles

is standard position, find two positive angles and two negative angles that are coterminal with

 to find positive coterminal angles, you can add any positive integer multiple of 360° to 

 to find negative coterminal angles, you can add any negative integer multiple  of 360° to

Radians


Definition: one radian is the measure of the central angle of a circle subtended by an arc equal in length to the radius of the circle.
The length of the radius is eqaul to the arc length.

Realtionships Between Degrees and Radians

1. 
2. 
3. 

Changing Angular Measures

Degrees to Radians: 

Radians to Degrees: 

Formula for the Length of a Circular Arc

If an arc of length s on a circle of radius r subtends a central angle or a radian measure θ, then s=rθ.

Formula for the Length of a Circular Sector

If θ is the radian measure of a central angle of a circle of radius r and if A is the area of the circular sector determined by θ, then A=1/2 r²θ.

-Austin A.K.A. the Hiphopopotamus