# 7. Solving Ax = 0: Pivot Variables, Special Solutions

- Channel: [MIT OpenCourseWare](https://www.youtube.com/channel/UCEBb1b_L6zDS3xTUrIALZOw)
- Published: 2009-05-07
- Source video: [Watch on YouTube](https://youtube.com/watch?v=VqP2tREMvt0)
- Read on ReadTube: https://www.read.tube/p/videos/VqP2tREMvt0
- Language: Original (source language)

## Summary

**Solving Ax = 0: Pivot Variables and Special Solutions**

This video details the algorithm for solving the equation Ax = 0 by utilizing Gaussian elimination to transform a matrix into its row echelon form. The process identifies pivot and free variables, enabling the calculation of special solutions that collectively define the entire null space.

To solve Ax = 0, perform Gaussian elimination on the matrix A to reach row echelon form (U). The number of pivots r determines the rank of the matrix, while the remaining n - r columns represent free variables. By assigning the value 1 to one free variable and 0 to all others, you can solve for the pivot variables using back substitution. This process yields special solutions that span the entire null space.

- The null space consists of all linear combinations of these special solutions.
- To find the special solutions systematically, reduce the matrix further to reduced row echelon form (R), which features an identity matrix in the pivot columns.
- The null space matrix N = [ -F; I ] captures all special solutions, where F is the free variable block and I is the identity matrix.

## Article

## Solving Ax = 0: Pivot Variables and Special Solutions

To understand the null space of a matrix—the collection of all vectors $x$ that satisfy the equation $Ax = 0$—we must move beyond definitions and establish a systematic algorithm. This process centers on Gaussian elimination, which allows us to find these solutions without changing the null space of the matrix.

### The Algorithm: From Elimination to Echelon Form

When solving $Ax = 0$, our goal is to perform row operations to reach an **echelon form** (also called staircase form). Throughout this process, we keep the right-hand side as zero. 

1.  **Elimination:** We perform row operations to reach an upper triangular-like structure. If we encounter a column with no pivot (a zero in the pivot position), we simply move to the next column.
2.  **Identifying Rank:** The number of pivots, denoted by $r$, is the **rank** of the matrix. This is the most crucial number in describing the matrix's properties.
3.  **Separating Variables:** 
    *   **Pivot Variables:** These correspond to columns containing a pivot.
    *   **Free Variables:** These correspond to columns without pivots. We can assign any values to these variables.

### Finding Special Solutions

Once we have our matrix in echelon form ($U$), we solve $Ux = 0$ using back substitution. Because we have $n - r$ free variables, we create "special solutions" by choosing convenient values for them:

*   Set one free variable to 1 and all other free variables to 0.
*   Solve for the pivot variables using back substitution.
*   Repeat this for every free variable.

The null space is the set of all linear combinations of these special solutions. If $x_p$ are the pivot variables and $x_f$ are the free variables, the solutions form a space defined by these vectors.

### The Reduced Row Echelon Form (RREF)

We can continue the elimination process further to reach the **reduced row echelon form (RREF)**, often denoted as $R$. This form is characterized by:

*   Pivots equal to 1.
*   Zeros above and below every pivot.
*   An identity matrix ($I$) occupying the pivot rows and columns.

In this form, the relationship becomes $Rx = 0$. If we partition $R$ into pivot columns and free columns (represented by a matrix $F$), the system looks like $[I F]x = 0$. The special solutions are then easily identified: the free variables are set to the identity matrix, and the pivot variables become $-F$. 

### Summary of Key Concepts

*   **Rank ($r$):** The number of independent rows/columns and the number of pivots.
*   **Free Variables:** Calculated as $n - r$ (total columns minus rank).
*   **Null Space:** The collection of all combinations of the special solutions. Each special solution corresponds to setting one free variable to 1 and the rest to 0.

By following this algorithm—eliminating to find the rank, identifying free variables, and setting them to create special solutions—we can completely describe the null space for any rectangular matrix.