Some Definitions

Nonzero Row or Column

A nonzero row or column in a matrix means a row or column that contains at least one nonzero entry.

Leading Entry

A leading entry of a row refers to the leftmost nonzero entry (in a nonzero row).

Echelon Form (阶梯形)

REF and RREF

A rectangular matrix is in row echelon form (REF) if it has the following properties:

1. All nonzero rows are above any rows of all zeros.
   • $\begin{bmatrix}0 & 2 & 4 \\ 3 & -1 & 0 \\ 0 & 0 & 0\end{bmatrix}$ satisfies this property.
   • $\begin{bmatrix}0 & 0 & 0 \\ 1 & 2 & 3 \\ 0 & 4 & 5\end{bmatrix}$ does not satisfy this property.
2. Each leading entry of a row is in a column to the right of the leading entry of the row above it.
   • $\begin{bmatrix}\color{blue}{1} & 2 & 0 & 3 \\ 0 & 0 & \color{blue}{4} & 5 \\ 0 & 0 & 0 & \color{blue}{6}\end{bmatrix}$ satisfies this property.
   • $\begin{bmatrix}0 & \color{blue}{2} & 3 \\ \color{red}{4} & 5 & 6 \\ 0 & 0 & \color{blue}{7}\end{bmatrix}$ does not satisfy this property.
3. All entries in a column below a leading entry are zeros.
   • In fact, any matrix that satisfies property 2 automatically satisfies this property.

If a matrix in echelon form satisfies the following additional conditions, then it is in reduced (row) echelon form (RREF):

4. The leading entry in each nonzero row is $1$.
5. Each leading $1$ is the only nonzero entry in its column.
   • $\begin{bmatrix}\color{blue}{1} & \color{red}{5} & 3 & 0 \\ 0 & \color{blue}{1} & 2 & 0 \\ 0 & 0 & 0 & \color{blue}{1} \\ 0 & 0 & 0 & 0\end{bmatrix}$ does NOT satisfies this property.

Theorem

Each matrix is row equivalent to one and only one reduced echelon matrix (RREF).

Pivot

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

Solutions of Linear Systems

Basic and Free Variables

Take the following linear system as an example:

$$\begin{bmatrix} 1 & 0 & -5 & 1 \\ 0 & 1 & 1 & 4 \\ 0 & 0 & 0 & 0 \end{bmatrix} \begin{cases} \begin{aligned} &x_1 & - 5&x_3 &= 1 \\ & &x_2 + &x_3 &= 4 \\ & & & 0 &= 0 \end{aligned} \end{cases}$$

• The variables $x_1$ and $x_2$ corresponding to the pivot columns in the matrix are called basic variables.
• The other variable, $x_3$, is called a free variable.

It is possible to rewrite the basic variables depending on the free variables:

$$\begin{cases} \begin{aligned} x_1 &= 1 + 5x_3 \\ x_2 &= 4 - x_3 \\ x_3 &\text{ is free} \end{aligned} \end{cases}$$

Consistence

A linear system is consistent if and only if the rightmost column of the augmented matrix is NOT a pivot column.

In other words, a linear system is inconsistent if and only if the augmented matrix has a row of the form $\begin{bmatrix}0 & 0 & \cdots & 0 & b\end{bmatrix}$, where $b$ is a nonzero number.

If a linear system is consistent, then it has either:

• a unique solution (if there are no free variables), or
• infinitely many solutions (if there is at least one free variable).

Copyright

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