Null Space
All solutions of Ax = 0.
Key idea
All solutions of Ax = 0.
In depth
The null space is the set of all x with Ax = 0. It is a subspace. Its dimension is the number of free variables. It measures what the transformation sends to zero.
Worked example
A = [1 2; 2 4]: x + 2y = 0 gives null space spanned by (−2, 1).
Formula
Real-world uses
- Finding hidden combinations that cancel out in circuits, structures and chemical balancing
Common mistake
Forgetting that the null space always contains 0.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Null Space?
All solutions of Ax = 0. It is a topic at the expert level in Linear Algebra.
What is the formula for Null Space?
The key formula is N(A) = {x : Ax = 0}.
Where is Null Space used in real life?
Real-world examples: Finding hidden combinations that cancel out in circuits, structures and chemical balancing.
What should I learn before Null Space?
Make sure you are comfortable with Column Space and Row Space, the previous topic in Linear Algebra.
What is a common mistake with Null Space?
Forgetting that the null space always contains 0.
How can I practise Null Space?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Null Space?
Move on to Rank next.
Related expert topics in Linear Algebra
- Triple Product – Scalar triple product gives parallelepiped volume.
- Homogeneous Systems – Ax = 0 always has the trivial solution.
- Cofactor Expansion – Compute determinants recursively.
- Properties of Determinants – Multiplicative; row swaps change sign.
- Geometric Meaning of Determinants – Area or volume scaling factor.
- Cramer’s Rule for Systems – Solve systems using determinants.