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