Basic Concepts in Linear Algebra

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

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

If a matrix in echelon form satisfies the following additional properties, it’s called reduced echelon form

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

• Pivot position. a location in matrix $A$ that corresponds to a leading $1$ in the reduced echelon form of $A$.
• Pivot column is a column of $A$ that contains a pivot position.