Kernel And Range

Definition

Linear Transformation

Map T: V → W between vector spaces V and W satisfying linearity: T(u + v) = T(u) + T(v), T(cv) = cT(v).

Kernel

Set of vectors in V mapped to zero vector in W: ker(T) = {v ∈ V | T(v) = 0}.

Range

Set of all vectors in W that are images of vectors in V: range(T) = {w ∈ W | w = T(v) for some v ∈ V}.

Kernel (Null Space)

Definition and Notation

Also called null space. Denoted ker(T) or null(T). Subspace of domain V.

Properties

• Subspace: closed under addition and scalar multiplication.
• Contains zero vector always.
• Measures failure of injectivity.

Interpretation

Vectors mapped to zero represent "loss" of information under T.

Range (Image)

Definition and Notation

Also called image. Denoted range(T) or im(T). Subspace of codomain W.

Properties

• Subspace of W.
• Contains all vectors attainable by T.
• Measures "reach" or coverage of transformation.

Interpretation

Describes outputs achievable by applying T to vectors in domain.

Properties of Kernel and Range

Subspace Structure

Both ker(T) ⊆ V, range(T) ⊆ W are vector subspaces.

Relation to Invertibility

Injective ⇔ ker(T) = {0}. Surjective ⇔ range(T) = W.

Closedness

Both are closed under vector addition and scalar multiplication.

Examples

Example 1: Zero Transformation

T(v) = 0 for all v ∈ V. Kernel = V, Range = {0}.

Example 2: Identity Transformation

T(v) = v. Kernel = {0}, Range = W = V.

Example 3: Projection onto Coordinate Axis

T: R² → R², T(x,y) = (x,0). Kernel = {(0,y)}, Range = {(x,0)}.

Dimension Theorem (Rank-Nullity)

Theorem Statement

For linear T: V → W with V finite-dimensional, dim(V) = dim(ker(T)) + dim(range(T)).

Rank and Nullity

Rank = dim(range(T)), Nullity = dim(ker(T)).

Implications

Controls dimension balance between kernel and image; key for solving linear systems.

dim(V) = rank(T) + nullity(T)

Injectivity and Surjectivity

Injective Maps

One-to-one: T(u) = T(v) ⇒ u = v. Equivalently, ker(T) = {0}.

Surjective Maps

Onto: range(T) = W.

Bijective Maps

Both injective and surjective; invertible linear transformations.

Computing Kernel and Range

Using Matrix Representation

Kernel: Solve homogeneous system Ax = 0. Range: Span of column vectors.

Gaussian Elimination

Reduce matrix to row echelon form to find solutions and pivot columns.

Algorithmic Steps

1. Write matrix A of T.2. Compute null space by solving Ax = 0.3. Identify pivot columns to form basis for range.4. Extract bases for ker(T), range(T). 

Matrix Representations

Kernel via Null Space of Matrix

ker(T) corresponds to null space of matrix A.

Range via Column Space

range(T) corresponds to column space of A.

Example Table

Applications

Solving Linear Systems

Kernel describes solutions to homogeneous system; range describes possible outputs.

Dimension Counting

Rank-nullity helps determine solvability and uniqueness.

Functional Analysis and Differential Equations

Kernel and range characterize operators and solution spaces.

Common Misconceptions

Kernel vs Range Confusion

Kernel is subset of domain; range is subset of codomain, not vice versa.

Zero Vector Inclusion

Kernel always contains zero vector; range may or may not.

Dimension Misinterpretation

Rank-nullity must be applied carefully only for finite-dimensional spaces.

Summary

Kernel: vectors mapped to zero; subspace of domain. Range: vectors attainable; subspace of codomain. Rank-nullity theorem links their dimensions. Key for understanding linear maps’ structure and behavior.

References

• Axler, S. "Linear Algebra Done Right." Springer, 3rd ed., 2015, pp. 45-70.
• Strang, G. "Introduction to Linear Algebra." Wellesley-Cambridge Press, 5th ed., 2016, pp. 120-150.
• Lang, S. "Linear Algebra." Springer-Verlag, 3rd ed., 1987, pp. 30-55.
• Halmos, P. R. "Finite-Dimensional Vector Spaces." Springer, 2nd ed., 1974, pp. 10-40.
• Roman, S. "Advanced Linear Algebra." Springer, 3rd ed., 2008, pp. 100-140.