Gauss-Jordan Elimination

Just like the Gaussian Elimination, Gauss- jordan elimination aims to solve a system of linear equations. This is also done by transforming the system into augmented matrix, but the difference is, it needs to be a reduced row-echelon form.

Reduced Row-Echelon Form
- is basically a simpler form of a matrix. A matrix is in reduced row-echelon form if:

1. If there is a row where every entry is zero, then this row lies below any other row that contains a nonzero entry.
2. The leftmost nonzero entry of a row is equal to 1.
3. The leftmost nonzero entry of a row is the only nonzero entry in its column.
4. The nonzero entry on the higher row should be further to the left than the nonzero entry on the lower row.

     See that the final augmented matrix, the nonzero entries are in a staircase pattern. That means the equivalent linear equations are x=1, y=4 and z=-2 because it's the only nonzero entries in that row.

    

Gaussian Elimination

Gaussian Elimination is performed to certain augmented matrices so that they can be in a row-echelon form. 

Steps
*From a System, transform it to an augmented matrix. 
*Perform Elementary Row Operations to transform it to Row-Echelon form.
*Do back-substitution.

3 Elementary Row Operations
1. Interchanging 2 rows.

2. Multiplying a row by a nonzero constant.
3. Adding a multiple of a row to another row. 

How to Know if the matrix is in Row-Echelon Form or not?

a. Rows with zero entries must be at the bottom of the matrix. 

b. All leading coefficients must be 1.
c. The leading coefficient of the higher row must be further to the left than the leading coefficient of the lower row.

What is Linear Algebra and What do i know about it?

Linear Algebra is the branch of mathematics concerning Vector spaces and linear mappings between such spaces. It is the study of lines, planes, and subspaces and their intersections using algebra.

List of Linear Algebra Topics:

Linear Equations
Matrices
Matrix Decompositions
Relations
Computations
Vector Spaces

What do i know about Linear equations?
        - Linear equations are collectively know as a system. Systems are composed of linear equations with the same set of variables. For example:









The above system is a system of three equations in three variables x,y, and z. A solution to a linear system is an assignment of numbers to the variables such that all equations are simultaneously satisfied.The solution to the above equation is x=1, y=-2, z=-2. 

What do i know about Matrices?
       -a matrix is a rectangular array of numbers, symbols, or expressions arranged in m rows and n columns. The individual items in a matrix are called entries. A matrix that has equal number of rows and columns (m=n) is called a square matrix. Matrices are linear systems and are solved using elementary row operations. 

Introduction to My Blog

Hi! I am Joel Ivan Sarmiento. This is my blog about Linear Algebra. I will try to explain things about Linear Algebra by posting lectures and notes about the subject. I will also have trivias and other stuffs to make it interactive. :D That's all Thank you.