Still Confused?

Try reviewing these fundamentals first

- Home
- Linear Algebra
- Linear Equations with Matrices

Still Confused?

Try reviewing these fundamentals first

Still Confused?

Try reviewing these fundamentals first

Nope, got it.

That's the last lesson

Start now and get better math marks!

Get Started NowStart now and get better math marks!

Get Started NowStart now and get better math marks!

Get Started NowStart now and get better math marks!

Get Started Now- Intro Lesson: a14:11
- Intro Lesson: b2:34
- Intro Lesson: c34:08
- Intro Lesson: d9:40
- Lesson: 1a4:48
- Lesson: 1b2:31
- Lesson: 1c2:50
- Lesson: 1d4:19
- Lesson: 2a23:46
- Lesson: 2b25:24
- Lesson: 3a14:57
- Lesson: 3b22:28
- Lesson: 4a10:38
- Lesson: 4b9:27

Basic Concepts:Solving a linear system with matrices using Gaussian elimination,

A **non-zero row** is a row that has at least one entry that is not zero.

The **leading entry** of a row is the leftmost non-zero entry in a non-zero row.

A rectangular matrix is in **echelon form** if it has the three properties:

- All non-zero rows are above any rows of all zeros.
- Each leading entry of a row is in a column to right of the leading entry of the row above it.
- All entries in a column below a leading entry are zeros.

If the rectangular matrix satisfies 2 more additional properties, then it is in **reduced echelon form**:

- The leading entry in each non-zero row is 1.
- Each leading 1 is the only non-zero entry in its column.

Essentially the difference between the two forms is that the reduced echelon form has 1 as a leading entry, and the column of the leading entry has 0's below and above.

The **pivot position** is just the **leading entries** of the echelon form matrix.

The **pivot column** is the column of the pivot position.

Here are the steps for the **row reduction algorithm**:

- Look for the leftmost non-zero column. This is our pivot column.
- Find a non-zero entry in the pivot column. This is our pivot position. It should be at the very top of the pivot column.
- Use matrix row operations to make all the entries below the pivot 0.
- Ignore the row with the pivot. Repeat Step 1-3 again and again until you can't anymore.
- Find your rightmost pivot. Make it 1. Then make all the entries above it 0.

- Introduction
**Row Reduction and Echelon Form Overview:**a)__Echelon Matrix vs. Reduced Echelon Matrix__- 3 properties for echelon form
- Addition 2 properties for reduced echelon form
- What is the difference between them?

b)__Pivot Position and Pivot Column__- pivot position = leading entry
- pivot column = column of pivot position

c)__The Row Reduction Algorithm__- The 5 steps of the algorithm
- Making sure it is in reduced echelon form

d)__Solutions of Linear Systems__- Reduced echelon form of augmented matrix
- Basic variables and free variables
- Writing out the solutions

- 1.
**Difference between Echelon Form and Reduced Echelon Form**

Label whether the following matrices are in echelon form or reduced echelon form:a)b)c)d) - 2.
**Learning the Row Reduction Algorithm**

Row reduce the matrices to reduced echelon form. Circle the pivot positions in the final and original matrix, and list the pivot columns from the original matrix in part b:a)b) - 3.
**Finding the General Solution of a Matrix**

Find the general solution of the following matrices:a)b) - 4.
**Linear Systems with Unknown Constants**

Choose values of a and b where the system has infinitely many solutions, and no solutions:a)$x_1+ax_2=4$

$2x_1+4x_2=b$b)$3x_1-x_2=6$

$x_1+ax_2=b$

1.

Linear Equations with Matrices

1.1

Notation of matrices

1.2

Solving systems of linear equations by graphing

1.3

Representing linear system as a matrix

1.4

The three types of matrix row operations

1.5

Solving a linear system with matrices using Gaussian elimination

1.6

Row reduction and echelon forms

1.7

Linear combination and vector equations in $R^n$

1.8

Matrix equation Ax=b

1.9

Solution sets of linear systems

1.10

Application of linear systems