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- Linear Equations with Matrices

Still Confused?

Try reviewing these fundamentals first

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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

Related 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