Span
All linear combinations of given vectors.
Key idea
All linear combinations of given vectors.
In depth
The span of vectors v₁, ..., vₖ is the set of all linear combinations c₁v₁ + ... + cₖvₖ. It is the smallest subspace containing them. Two non-parallel vectors in the plane span all of it.
Worked example
Span{(1, 0), (0, 1)} = R². Span{(1, 2)} is a line.
Real-world uses
- Finding every combination of ingredients or colours that a set of paints can produce
Common mistake
Believing that more vectors always span a larger space.
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 Span?
All linear combinations of given vectors. It is a topic at the expert level in Linear Algebra.
Where is Span used in real life?
Real-world examples: Finding every combination of ingredients or colours that a set of paints can produce.
What should I learn before Span?
Make sure you are comfortable with Subspaces, the previous topic in Linear Algebra.
What is a common mistake with Span?
Believing that more vectors always span a larger space.
How can I practise Span?
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 Span?
Move on to Linear Independence 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.