Kernel & Range
Linear Transformations
Discover what maps to zero and what the transformation can reach: the two fundamental subspaces that reveal the structure of every linear map.
What is a Linear Transformation?
A linear transformation \( T: V \to W \) is a function between vector spaces that preserves the two core operations of linear algebra:
Defining the Kernel
The kernel (or null space) of a linear transformation \( T: V \to W \) is the set of all vectors in \( V \) that map to the zero vector in \( W \):
Every vector in the kernel "collapses" to \( \mathbf{0} \). The kernel always contains at least \( \mathbf{0}_V \) (since \( T(\mathbf{0}) = \mathbf{0} \)).
Defining the Range (Image)
The range (or image) of \( T: V \to W \) is the set of all vectors in \( W \) that are actually "hit" by the transformation:
The range might be all of \( W \) (in which case \( T \) is surjective / onto), or it might be a proper subspace of \( W \).
Both are Subspaces
A key fact: the kernel and range are not just arbitrary sets. They are subspaces of their respective vector spaces.
The Rank-Nullity Theorem
The most important theorem connecting kernel and range relates their dimensions:
Rank-Nullity Visualized
Kernel vs Range: Key Distinction
| Kernel (Null Space) | Range (Image) | |
|---|---|---|
| Definition | \( \{ \mathbf{v} \mid T(\mathbf{v}) = \mathbf{0} \} \) | \( \{ T(\mathbf{v}) \mid \mathbf{v} \in V \} \) |
| Lives in | Domain \( V \) | Codomain \( W \) |
| Dimension | Nullity | Rank |
| Tells us about | Information lost | Information preserved |
| Injectivity | Injective iff \( \ker T = \{\mathbf{0}\} \) | Surjective iff Range \( = W \) |
How to Find Kernel & Range
Example 1: A Simple Matrix
Let \( T: \mathbb{R}^3 \to \mathbb{R}^2 \) be defined by the matrix:
Step 1: Row reduce. \( A \) is already close to RREF. Subtract \( 2R_2 \) from \( R_1 \):
Step 2: Kernel. Set the free variable \( x_3 = t \). Then \( x_1 = t,\; x_2 = -t \):
Example 1: Range & Verification
Step 3: Range. Pivot columns of \( A \) are columns 1 and 2:
Step 4: Verify Rank-Nullity.
Example 2: A Projection
Let \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) project onto the \( xy \)-plane: \( T(x, y, z) = (x, y, 0) \).
Example 3: A Harder Case
Let \( T: \mathbb{R}^4 \to \mathbb{R}^3 \) with:
Pivots in columns 1, 2. Free variables: \( x_3 = s,\; x_4 = t \).
Common Mistakes to Avoid
Summary Reference Table
| Property | Kernel | Range |
|---|---|---|
| Notation | \( \ker(T) \), \( \text{Null}(A) \) | \( \text{Range}(T) \), \( \text{Col}(A) \) |
| Subspace of | \( V \) (domain) | \( W \) (codomain) |
| Dimension called | Nullity | Rank |
| How to find | Solve \( A\mathbf{x} = \mathbf{0} \) | Pivot columns of \( A \) |
| Key theorem | \( \text{nullity} + \text{rank} = n \) (Rank-Nullity) | |
| Injectivity | \( T \) injective iff trivial | \( T \) surjective iff \( = W \) |
Question 1 of 5
The kernel of a linear transformation \( T: V \to W \) is a subspace of which vector space?
Question 2 of 5
If \( A \) is a \( 3 \times 5 \) matrix with rank 3, what is the nullity?
Question 3 of 5
A linear transformation \( T \) is injective (one-to-one) if and only if:
Question 4 of 5
Given \( A = \begin{pmatrix} 1 & 2 \\ 3 & 6 \end{pmatrix} \), which vector is in \( \ker(A) \)?
Question 5 of 5
When finding a basis for the range (column space) of a matrix \( A \), you should use the pivot columns from: