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

3Blue1Brown

3Blue1Brown

9 min, 59 sec

The video delves into the concepts of vector coordinates, linear combinations, basis vectors, and spans in linear algebra.

Summary

  • The video explains vector coordinates as scalars that stretch or squish basis vectors i-hat and j-hat in the xy coordinate system.
  • Reiterates the concept of adding together two scaled vectors and introduces basis vectors as a special name for i-hat and j-hat.
  • Discusses how changing the basis vectors leads to a different coordinate system and how every two-dimensional vector can be represented through a new set of basis vectors.
  • Explains the term 'linear combination' as the addition of scaled vectors and probes into the etymology related to lines.
  • Explores the span of vectors in both two-dimensional and three-dimensional spaces, along with the concepts of linear dependence and independence.

Chapter 1

Recap and Introduction to Vector Coordinates

0:11 - 18 sec

Recapping vector addition and scalar multiplication, and introducing vector coordinates as scalars.

Recapping vector addition and scalar multiplication, and introducing vector coordinates as scalars.

  • Recalls previous discussions on vector addition and scalar multiplication.
  • Presents vector coordinates, translating between pairs of numbers and two-dimensional vectors.
  • Introduces the concept of coordinates as scalars affecting basis vectors.

Chapter 2

Basis Vectors in the Coordinate System

0:45 - 35 sec

Explaining the role of basis vectors i-hat and j-hat in the xy coordinate system.

Explaining the role of basis vectors i-hat and j-hat in the xy coordinate system.

  • Identifies i-hat and j-hat as basis vectors in the xy coordinate system.
  • Illustrates how the x and y coordinates scale these basis vectors.
  • Emphasizes the significance of adding scaled vectors.

Chapter 3

Flexibility of Choosing Basis Vectors

1:27 - 1 min, 17 sec

Discussing the flexibility and implications of choosing different basis vectors for coordinate systems.

Discussing the flexibility and implications of choosing different basis vectors for coordinate systems.

  • Highlights the possibility of adopting different basis vectors for new coordinate systems.
  • Explains how different vectors can be reached by varying scalars.
  • Encourages contemplation on the concept of reaching all two-dimensional vectors.

Chapter 4

Linear Combinations and the Concept of Span

3:02 - 2 min, 28 sec

Defining linear combinations, introducing the concept of span, and discussing vector representation.

Defining linear combinations, introducing the concept of span, and discussing vector representation.

  • Defines 'linear combination' as scaling and adding vectors.
  • Introduces the term 'span' for the set of all reach vectors through linear combinations.
  • Differentiates between thinking of vectors as arrows versus points.

Chapter 5

Span in Three-Dimensional Space

5:58 - 2 min, 23 sec

Expanding the concept of span to three-dimensional vectors.

Expanding the concept of span to three-dimensional vectors.

  • Illustrates the span of two vectors in three-dimensional space.
  • Describes how adding a third vector can affect the span.
  • Explains the span of three vectors in terms of accessing all three dimensions of space.

Chapter 6

Linear Dependence and Independence

8:30 - 56 sec

Clarifying linear dependence and independence among vectors.

Clarifying linear dependence and independence among vectors.

  • Defines 'linearly dependent' vectors as ones that don't expand the span.
  • States 'linearly independent' vectors add dimensions to the span.
  • Poses a puzzle to understand the technical definition of a basis.

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