The Null Space (Kernel)

The World of Lost Information

In our last lesson, we focused on the **Column Space**—the set of all possible *outputs* `b`. Now, we ask a fundamentally different question. Let's consider the simplest possible target: the zero vector, `0`.

The set of **all possible solutions `x`** to the homogeneous equation `Ax = 0` is called the **Null Space** of the matrix `A`.

The Null Space

\text{Null Space } N(A) = \{ \text{all vectors } x \text{ such that } Ax = 0 \}

The Null Space is the set of all input vectors that get **squashed, collapsed, or annihilated** into the single point of the origin by the transformation `A`. It is always a **subspace** of the input space.

Finding a Basis for the Null Space

The Null Space is defined by the solutions to `Ax=0`. How do we find these solutions? With the most powerful tool in our toolbox: **Gauss-Jordan Elimination (to RREF)**.

Step 1: Reduce `A` to its Reduced Row Echelon Form (RREF)

For `Ax=0`, we can just focus on `A` because the zero vector on the right side of an augmented matrix never changes.

A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 4 & 7 & 9 \\ 3 & 6 & 8 & 11 \end{bmatrix} \quad \xrightarrow{\text{RREF}} \quad R = \begin{bmatrix} \mathbf{1} & 2 & 0 & 1 \\ 0 & 0 & \mathbf{1} & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix}

Step 2: Identify pivot variables and free variables

Pivot Columns: 1 and 3. So, `x₁` and `x₃` are pivot variables.
Free Columns: 2 and 4. So, `x₂` and `x₄` are **free variables**.

Step 3: Write the general solution to `Rx = 0`

Let `x₂ = s` and `x₄ = t`, where `s` and `t` can be any real numbers.

From Row 1: $x₁ + 2x₂ + x₄ = 0 \implies x₁ = -2s - t$

From Row 2: $x₃ + x₄ = 0 \implies x₃ = -t$

Step 4: Decompose the solution into a linear combination of vectors

This is the magic step. We write our solution vector `x` and separate the parts for `s` and `t`.

x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix} -2s - t \\ s \\ -t \\ t \end{bmatrix} = s \begin{bmatrix} -2 \\ 1 \\ 0 \\ 0 \end{bmatrix} + t \begin{bmatrix} -1 \\ 0 \\ -1 \\ 1 \end{bmatrix}

Every vector `x` in the Null Space is a linear combination of these two specific vectors. These two vectors form a **basis for the Null Space of `A`**.

The dimension of the Null Space (called the **nullity**) is 2, which is exactly the number of free variables we had.

The Meaning of the Null Space: Redundancy

The Null Space is the precise mathematical description of **redundancy** in the columns of `A`.

A non-zero vector `x` in the Null Space means that `Ax = 0`, which is `x₁·(col 1) + x₂·(col 2) + ... = 0`. This is the very definition of **linear dependence** among the columns of `A`. For `s=1, t=0`, we find `col 2 = 2 * col 1`.

Lesson 2.6: Column Space & Rank

Lesson 2.8: Row Space & Left Null Space