# Vector Independence, Span, Basis, and Dimension

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Lesson Summary:

In this lecture, we learn about vector independence, span, basis, and dimension. We are introduced to the definitions of linear independence and span, and we see how they relate to each other. A set of vectors is independent if no combination of them gives the zero vector, and they span a space if all their linear combinations fill up the entire space. The concept of a basis is also introduced, which is a set of vectors that is both independent and spans the space. The lecture includes examples in two and three-dimensional space to demonstrate these concepts.

Lesson Description:

Vector Independence, Span, Basis, and Dimension -- Lecture 9. Learn when vectors are independent or dependent, what a vector span is, what a basis is, and how to find the dimension.

Gilbert Strang, 18.06 Linear Algebra, Spring 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed November 19, 2008). License: Creative Commons BY-NC-SA.

• What does it mean for vectors to be independent?
• When are vectors dependent?
• What does it mean for vectors to span a space?
• What is a basis for a space?
• How do you find the dimension in linear algebra?
• #### Staff Review

• Currently 4.0/5 Stars.
This video is a great video for learning about vector independence and dependence, and how to know whether vectors are independent or dependent. It also discusses what a vector span is, a basis, and how to find their dimension. A very diversified and full lesson. There is a small error in this video that is corrected in lecture 10.