Basis of a subspace | Vectors and spaces | Linear Algebra | Khan Academy

TL;DR

Subspaces in linear algebra are defined by the span of a set of vectors, and a basis is a set of linearly independent vectors that span a subspace.

Transcript

Let's say I have the subspace v. And this is a subspace and we learned all about subspaces in the last video. And it's equal to the span of some set of vectors. And I showed in that video that the span of any set of vectors is a valid subspace. It's going to be the span of v1, v2, all the way, so it's going to be n vectors. So each of these are vec... Read More

Key Insights

• 😫 A subspace in linear algebra is defined as the span of a set of vectors.
• 😫 A basis for a subspace is a set of linearly independent vectors that span the subspace.
• 😫 A basis is the minimum set of vectors needed to span a subspace, with no redundancy.
• ❓ Subspaces can have multiple bases.

Questions & Answers

Q: What is the definition of a subspace?

A subspace is a set of vectors that can be represented as the span of a set of vectors, meaning it includes all possible linear combinations of those vectors.

Q: What does it mean for vectors to be linearly independent?

Vectors are linearly independent if the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0 is when all the coefficients c1, c2, ..., cn are equal to 0.

Q: Can a subspace have multiple bases?

Yes, a subspace can have multiple bases. As long as the vectors in the basis are linearly independent and span the subspace, they can be considered a basis.

Q: How can a vector in a subspace be uniquely defined by a basis?

Any vector in a subspace can be represented as a unique combination of the vectors in its basis. This means that each vector in the subspace can be expressed as a linear combination of the basis vectors with unique coefficients.

Summary & Key Takeaways

• A subspace in linear algebra is defined as the span of a set of vectors.

• In order for a set of vectors to be a basis for a subspace, they need to be linearly independent and span the subspace.

• A basis is the minimum set of vectors needed to span a subspace, with no redundancy.