Matrices Elimination

Matrices elimination (or solving system of linear equations) is the very first and fundamental skill throughout Linear Algebra. It's probably the first lesson of all sorts of courses.

Terminology

Before learning solving systems of linear equations, you really need to get familiar with all the core terminologies involved, otherwise it can be very hard to move on to next stage.
And in this case, the best way to learn that is through Wikipedia.

JFR, the core terms are: Gaussian elimination, Gauss-Jordan elimination, Augmented Matrix, Elementary Row Operations, Elementary matrix, Row Echelon Form (REF), Reduced Row Echelon Form (RREF), Triangular Form.

Gaussian elimination

It’s a Row reduction algorithm to solve System of linear equations.

Refer to Wiki: Gaussian elimination
Refer to simple wiki: Gaussian elimination
Example: showme.com

To perform Gaussian elimination, the coefficients of the terms in the system of linear equations are used to create a type of matrix called an augmented matrix.
Then, elementary row operations are used to simplify the matrix.
The goal of Gaussian elimination is to get the matrix in row-echelon form.
If a matrix is in row-echelon form, which is also called Triangular Form.
Some definitions of Gaussian elimination say that the matrix result has to be in reduced row-echelon form.
Gaussian elimination that creates a reduced row-echelon matrix result is sometimes called Gauss-Jordan elimination.

To be simpler, here is the structure:

• Algorithm: Gaussian Elimination
• Step 1: Rewrite system to a Augmented Matrix.
• Step 2: Simplify matrix with Elementary row operations.
• Result:
• Row Echelon Form or
• Reduced Echelon Form

And if we make the result only in RREF, so the name of the algorithm could also be called:

• Algorithm: Gauss-Jordan Elimination
• Step 1: Rewrite system to a Augmented Matrix.
• Step 2: Simplify matrix with Elementary row operations.
• Result: Only in Reduced Echelon Form

Elementary Row Operations

Elementary row operations are used to simplify the matrix.

The three types of row operations used are:

• Type 1: Switching one row with another row.
• Type 2: Multiplying a row by a non-zero number.
• Type 3: Adding a row from another row. (!Note: you can only ADD them but not subtract, but you can ADD a negative)

Confusing operation: See where the negative sign was put:

Example

Suppose the goal is to find the solution for the linear system below:

First we need to turn it into Augmented Matrix form:

Then we apply Elementary Row Operations, and result in Row Echelon Form:

At the end, if we’d like, we can further on apply some row operations to get the matrix in Reduced Row Echelon Form:

Reading this matrix tells us that the solutions for this system of equations occur when x = 2, y = 3, and z = -1.

Row Echelon Form vs. Reduced Row Echelon Form

Refer to this lecture video: REF & RREF.

It doesn’t really matter it is a Square Matrix or not, there could be a Diagonal or Main diagonal, or you can't draw a diagonal at all.
The only thing matters is WHAT ARE ABOVE 1 AND WHAT ARE BELOW 1.

• REF: For each column, all numbers below 1 MUST BE 0. Doesn’t matter what numbers are above 1.
• RREF: For each column, all numbers both above & below 1 MUST BE 0. We don’t care about it if there’s no 1 in the column.

Augmented Matrix

Means we put another column into the matrix, which represents the Right side of the system of equations, numbers of right of the = sign.

When we apply elimination to Linear equations, we operate both sides at the same time. But for computer programmes, it often apply to Left side, and remember the operations, a.g. multiply a number or add equations together, when the left side finished then apply the same operations to the right side.

If a given Matrix was told it’s an Augmented Matrix, so we have to assume that the Last Column is The Solution Column.

Equivalent systems & Equivalent Matrices

• Equivalent systems: Linear systems with the SAME SOLUTION SET.

• Equivalent matrices: Two matrices where One Matrix can be turned into the other matrix by some elementary row operations.

Pivot

Or called the Cursor, or Basic, or Basic variable.

Refer to this video from mathispower4u.

It means the value that represents the unknown variable in each column. There's no pivot in a column if you can't get a 1 in that column.

Free variables

If there’s no pivot in a column, that means this unknown variable of the column can be any number, so we call it a free variable.

Pivot columns

The pivots are found after Row Reduction, and then go back to the Original Matrix, the columns WITH pivots are called pivot columns.

Back Substitution

It’s simple: When you solve out one unknown variable in the Linear System, you put the value back to other equations. We call this process as Back Substitution.

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