Null Space and Kernel

Definition

The null space (or kernel) of a linear transformation $T: V \rightarrow W$ is the set of all vectors in $V$ that map to the zero vector in $W$:

where $\vec{0}_W$ is the zero vector in $W$.

Properties

1. Subspace: The null space of a linear transformation $T: V \rightarrow W$ is a subspace of $V$.

2. Dimension: The dimension of the null space is called the nullity of $T$:

3. Injectivity: A linear transformation $T$ is injective (one-to-one) if and only if $\ker(T) = {\vec{0}_V}$, i.e., the null space contains only the zero vector.
   • This means distinct inputs always yield distinct outputs
   • By the Rank-Nullity Theorem, $T$ can only be injective if $\dim(V) \leq \dim(W)$

Null Space of a Matrix

For a matrix $A \in \mathbb{R}^{m \times n}$, the null space of $A$ is the set of all vectors $\vec{x} \in \mathbb{R}^n$ such that $A\vec{x} = \vec{0}$:

This corresponds to the null space of the linear transformation $T_A: \mathbb{R}^n \rightarrow \mathbb{R}^m$ defined by $T_A(\vec{x}) = A\vec{x}$.

Finding the Null Space of a Matrix

To find a basis for the null space of a matrix $A$:

1. Transform $A$ into row-reduced echelon form (RREF) using Gaussian elimination.
2. Identify the free variables (variables that are not leading variables in the RREF).
3. Express the leading variables in terms of the free variables.
4. Write the general solution as a linear combination of vectors, each corresponding to setting one free variable to 1 and the others to 0.
5. These vectors form a basis for the null space.

Example

Consider the matrix:

The RREF of $A$ is:

The equation $A\vec{x} = \vec{0}$ becomes:

This gives us:

Solving for $x_1$:

The general solution is:

Therefore, a basis for the null space of $A$ is:

And the nullity of $A$ is 2.

Rank-Nullity Theorem

The Rank-Nullity Theorem establishes a fundamental relationship between the null space and image:

Theorem: If $T: V \rightarrow W$ is a linear transformation and $V$ is finite-dimensional, then:

where $\text{rank}(T) = \dim(\text{Im}(T))$ and $\text{nullity}(T) = \dim(\ker(T))$.

This theorem provides powerful insights:

• The sum of the dimensions of the image and kernel equals the dimension of the domain
• Understanding one space helps characterize the other
• It directly connects to the concepts of injectivity and surjectivity

Interpretation

The Rank-Nullity Theorem tells us that the dimension of the domain equals the sum of:

• The dimension of the image (how much of the domain is "preserved" by the transformation)
• The dimension of the kernel (how much of the domain is "collapsed" to zero)

Matrix Form

For a matrix $A \in \mathbb{R}^{m \times n}$:

Applications

1. Systems of Linear Equations: The null space of a coefficient matrix represents the solution space of the homogeneous system.

2. Linear Independence: Vectors ${v_1, v_2, \ldots, v_k}$ are linearly independent if and only if the null space of the matrix with these vectors as columns contains only the zero vector.

3. Eigenvalues and Eigenvectors: For a square matrix $A$, the eigenvectors corresponding to eigenvalue $\lambda$ form the null space of $(A - \lambda I)$.

4. Differential Equations: The null space of a differential operator corresponds to the solution space of the homogeneous differential equation.

Exercise

Find a basis for the null space of the matrix:

Verify the Rank-Nullity Theorem for this matrix.