Linear Independence Span and Basis in Linear Algebra
Linear independence, span, and basis are the foundational vocabulary of vector spaces, tested heavily in university linear algebra courses and on exams like MIT 18.06. This set covers the definitions of each concept, how to test independence using row reduction, and how basis size determines subspace dimension — building blocks for eigenspaces, column spaces, and null spaces.
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5 CardsHow to test linear independence
What is the dimension of a vector space?
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What is the difference between span and basis?
The span of a set is all linear combinations of those vectors — it may include redundant vectors. A basis is a minimal spanning set: linearly independent vectors that span the space.
- Span: focuses on reachability
- Basis: adds the constraint of independence
How many vectors can be in a basis for Rⁿ?
Any basis for Rⁿ contains exactly n vectors. Adding more creates linear dependence; removing any loses the spanning property. This count is the dimension of Rⁿ.
Why is linear independence important?
Linear independence ensures a set of vectors has no redundancy — each vector contributes a new direction. It is essential for defining bases, solving systems uniquely, and computing rank and subspace dimension.