Basis
A linearly independent spanning set.
Key idea
A linearly independent spanning set.
In depth
A basis is an independent set that spans the space. Every vector has a unique representation in a basis. It is the smallest spanning set and the largest independent set.
Worked example
{(1, 0), (0, 1)} and {(1, 1), (1, −1)} are both bases of R².
Real-world uses
- Choosing coordinate axes for a robot, a dataset or a satellite image in space
Common mistake
Believing a space has just one basis.
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 Basis?
A linearly independent spanning set. It is a topic at the expert level in Linear Algebra.
Where is Basis used in real life?
Real-world examples: Choosing coordinate axes for a robot, a dataset or a satellite image in space.
What should I learn before Basis?
Make sure you are comfortable with Linear Independence, the previous topic in Linear Algebra.
What is a common mistake with Basis?
Believing a space has just one basis.
How can I practise Basis?
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 Basis?
Move on to Dimension 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.