Dimension
The number of vectors in a basis.
Key idea
The number of vectors in a basis.
In depth
The dimension is the number of vectors in any basis. It is well defined because all bases of a space have the same size. It tells how many free parameters are needed to describe a vector.
Worked example
R³ has dimension 3. The plane x + y + z = 0 has dimension 2.
Real-world uses
- Estimating how many independent factors explain a dataset in statistics and machine learning
Common mistake
Counting the dimension of the ambient space instead of the subspace.
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 Dimension?
The number of vectors in a basis. It is a topic at the expert level in Linear Algebra.
Where is Dimension used in real life?
Real-world examples: Estimating how many independent factors explain a dataset in statistics and machine learning.
What should I learn before Dimension?
Make sure you are comfortable with Basis, the previous topic in Linear Algebra.
What is a common mistake with Dimension?
Counting the dimension of the ambient space instead of the subspace.
How can I practise Dimension?
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 Dimension?
Move on to Coordinates Relative to a Basis 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.