Rank-Nullity Theorem Calculator

Formula first

Overview

In the context of a linear map T: V → W where V is finite-dimensional, this theorem provides a fundamental constraint on the relationship between the dimensions of the kernel and the image.

Symbols

Variables

$\dim$(V) = Dimension of Domain, $\text{rank}$(T) = Rank, $\text{nullity}$(T) = Nullity

Apply it well

When To Use

When to use: This theorem is the most fundamental tool in undergraduate linear algebra for determining the dimensions of subspaces associated with linear transformations.

Why it matters: It links the concept of injectivity (connected to the nullity) and surjectivity (connected to the rank) to the geometry of the domain space.

Avoid these traps

Common Mistakes

• Confusing the dimension of the codomain (W) with the dimension of the domain (V).
• Assuming the theorem applies to non-linear transformations.

One free problem

Practice Problem

Given a linear transformation T: ℳ → Ⅎ where the kernel is a line through the origin (dimension 1), calculate the rank of T.

Solve for: $x$

Hint: The dimension of the domain is 3. If the nullity is 1, use the theorem: Rank + Nullity = Dim(V).

The full worked solution stays in the interactive walkthrough.

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Related Formulas

References

Sources

1. Axler, S. (2015). Linear Algebra Done Right.
2. Axler, Sheldon. Linear Algebra Done Right. Springer, 3rd ed., 2015.
3. Strang, Gilbert. Introduction to Linear Algebra. Wellesley-Cambridge Press, 5th ed., 2016.
4. Wikipedia: Rank-nullity theorem
5. Rank-nullity theorem (Wikipedia article)
6. Sheldon Axler Linear Algebra Done Right
7. Gilbert Strang Introduction to Linear Algebra
8. Wikipedia article 'Rank-nullity theorem'