Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra

About this video

• Video Title: Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra
• Channel: 3Blue1Brown
• Speakers: Grant Sanderson
• Duration: 00:09:59

Overview

This video introduces the concepts of linear combinations, span, and basis vectors in the context of linear algebra. It explains how coordinates can be interpreted as scalars that scale basis vectors, and how the sum of these scaled vectors forms a linear combination. The video further explores the geometric interpretation of the span of vectors as the set of all reachable vectors through linear combinations, illustrating this in 2D and 3D space. Finally, it defines linear dependence and independence and introduces the technical definition of a basis.

Key takeaways

• Linear Combination: A vector can be represented as the sum of scaled basis vectors, where the scalars are the coordinates. For example, the vector (3, -2) can be seen as 3 times the x-basis vector (i-hat) plus -2 times the y-basis vector (j-hat).
• Basis Vectors: These are fundamental vectors that define a coordinate system. The standard basis vectors in a 2D xy-coordinate system are i-hat (unit vector in x-direction) and j-hat (unit vector in y-direction). The choice of basis vectors can vary, leading to different coordinate systems.
• Span: The span of a set of vectors is the set of all possible vectors that can be created by taking linear combinations of those vectors. In 2D, the span of two non-collinear vectors is the entire plane, while the span of two collinear vectors is a line. In 3D, the span of two non-collinear vectors is a plane through the origin, and the span of three non-coplanar vectors is the entire 3D space.
• Linear Dependence and Independence: A set of vectors is linearly dependent if one vector can be expressed as a linear combination of the others (i.e., it's redundant and doesn't expand the span). If each vector adds a new dimension to the span, the set is linearly independent.
• Basis (Technical Definition): A basis of a space is a set of linearly independent vectors that span that space. This means they are the minimal set of vectors needed to reach every point in the space.