Subspaces
Subsets that are themselves vector spaces.
Key idea
Subsets that are themselves vector spaces.
In depth
A subspace is a subset of a vector space that is itself a vector space. Test: it contains 0, is closed under addition, and closed under scalar multiplication. Lines and planes through the origin are subspaces.
Worked example
The plane x + y + z = 0 is a subspace of R³. The plane x + y + z = 1 is not.
Real-world uses
- Describing the solutions of a system as a plane or a line inside 3D space
Common mistake
Forgetting that a subspace must contain the zero vector.
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 Subspaces?
Subsets that are themselves vector spaces. It is a topic at the expert level in Linear Algebra.
Where is Subspaces used in real life?
Real-world examples: Describing the solutions of a system as a plane or a line inside 3D space.
What should I learn before Subspaces?
Make sure you are comfortable with Vector Spaces, the previous topic in Linear Algebra.
What is a common mistake with Subspaces?
Forgetting that a subspace must contain the zero vector.
How can I practise Subspaces?
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 Subspaces?
Move on to Span 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.