Matrix multiplication as composition | Chapter 4, Essence of linear algebra
3Blue1Brown
10 min, 4 sec
The video explains linear transformations, matrix representation of these transformations, and the mechanics and geometric interpretation of matrix multiplication, including the composition of transformations.
Summary
- Linear transformations are functions that take vectors to vectors, visually represented by space manipulation where grid lines remain parallel, evenly spaced, and the origin is fixed.
- A transformation's effect on a vector is determined by its effect on the basis vectors, leading to matrix representation of linear transformations.
- Matrix multiplication is the computational application of a transformation, representing a new transformation resulting from applying one transformation after another.
- The video demonstrates how to find the matrix representing the composition of two transformations and discusses the non-commutative property of matrix multiplication.
- Associativity of matrix multiplication is explained conceptually by the sequencing of transformations.
Chapter 1
![The video begins with a recap of linear transformations and their matrix representations.](https://www.videogist.co/rails/active_storage/representations/redirect/eyJfcmFpbHMiOnsiZGF0YSI6MTY2NjgzLCJwdXIiOiJibG9iX2lkIn19--df81c1a9f678f42d207a1f337badac9b01716dd9/eyJfcmFpbHMiOnsiZGF0YSI6eyJmb3JtYXQiOiJqcGciLCJyZXNpemVfdG9fbGltaXQiOls3MjAsbnVsbF19LCJwdXIiOiJ2YXJpYXRpb24ifX0=--c9426325207613fdd890ee7713353fad711030c7/9673_62.jpg)
The video begins with a recap of linear transformations and their matrix representations.
- Linear transformations are functions that map vectors to other vectors while keeping grid lines parallel and evenly spaced.
- The transformation of a vector can be determined by its effect on the basis vectors i-hat and j-hat.
- The coordinates of transformed i-hat and j-hat are recorded as columns of a matrix, defining matrix-vector multiplication.
![The video begins with a recap of linear transformations and their matrix representations.](https://www.videogist.co/rails/active_storage/representations/redirect/eyJfcmFpbHMiOnsiZGF0YSI6MTY2NjgzLCJwdXIiOiJibG9iX2lkIn19--df81c1a9f678f42d207a1f337badac9b01716dd9/eyJfcmFpbHMiOnsiZGF0YSI6eyJmb3JtYXQiOiJqcGciLCJyZXNpemVfdG9fbGltaXQiOls3MjAsbnVsbF19LCJwdXIiOiJ2YXJpYXRpb24ifX0=--c9426325207613fdd890ee7713353fad711030c7/9673_62.jpg)
Chapter 2
![The concept of composition of transformations is introduced.](https://www.videogist.co/rails/active_storage/representations/redirect/eyJfcmFpbHMiOnsiZGF0YSI6MTY2Njg3LCJwdXIiOiJibG9iX2lkIn19--7cab30363809e3ada80458313f10d7e7602103cb/eyJfcmFpbHMiOnsiZGF0YSI6eyJmb3JtYXQiOiJqcGciLCJyZXNpemVfdG9fbGltaXQiOls3MjAsbnVsbF19LCJwdXIiOiJ2YXJpYXRpb24ifX0=--c9426325207613fdd890ee7713353fad711030c7/9673_148.jpg)
The concept of composition of transformations is introduced.
- Applying one transformation after another creates a new linear transformation, described as the composition of the two.
- The composition has its own matrix, which can be deduced by tracking the movement of i-hat and j-hat through both transformations.
- The new matrix captures the combined effect of the transformations.
![The concept of composition of transformations is introduced.](https://www.videogist.co/rails/active_storage/representations/redirect/eyJfcmFpbHMiOnsiZGF0YSI6MTY2Njg3LCJwdXIiOiJibG9iX2lkIn19--7cab30363809e3ada80458313f10d7e7602103cb/eyJfcmFpbHMiOnsiZGF0YSI6eyJmb3JtYXQiOiJqcGciLCJyZXNpemVfdG9fbGltaXQiOls3MjAsbnVsbF19LCJwdXIiOiJ2YXJpYXRpb24ifX0=--c9426325207613fdd890ee7713353fad711030c7/9673_148.jpg)
Chapter 3
![The geometric meaning of matrix multiplication is emphasized.](https://www.videogist.co/rails/active_storage/representations/redirect/eyJfcmFpbHMiOnsiZGF0YSI6MTY2Njg1LCJwdXIiOiJibG9iX2lkIn19--8444a9971cb4a23a7b4e6df429e907a7b8b26474/eyJfcmFpbHMiOnsiZGF0YSI6eyJmb3JtYXQiOiJqcGciLCJyZXNpemVfdG9fbGltaXQiOls3MjAsbnVsbF19LCJwdXIiOiJ2YXJpYXRpb24ifX0=--c9426325207613fdd890ee7713353fad711030c7/9673_212.jpg)
The geometric meaning of matrix multiplication is emphasized.
- Matrix multiplication is numerically equivalent to applying subsequent transformations.
- The composition matrix of two transformations can be found by applying transformations to vectors and using the results as columns in the matrix.
- The order of matrix multiplication matters, as it reflects the sequence of transformations applied.
![The geometric meaning of matrix multiplication is emphasized.](https://www.videogist.co/rails/active_storage/representations/redirect/eyJfcmFpbHMiOnsiZGF0YSI6MTY2Njg1LCJwdXIiOiJibG9iX2lkIn19--8444a9971cb4a23a7b4e6df429e907a7b8b26474/eyJfcmFpbHMiOnsiZGF0YSI6eyJmb3JtYXQiOiJqcGciLCJyZXNpemVfdG9fbGltaXQiOls3MjAsbnVsbF19LCJwdXIiOiJ2YXJpYXRpb24ifX0=--c9426325207613fdd890ee7713353fad711030c7/9673_212.jpg)
Chapter 4
![An example is provided to illustrate how to compute a composition matrix.](https://www.videogist.co/rails/active_storage/representations/redirect/eyJfcmFpbHMiOnsiZGF0YSI6MTY2Njg5LCJwdXIiOiJibG9iX2lkIn19--85d1ff435f979cc3a68dd1135029a91e552cce38/eyJfcmFpbHMiOnsiZGF0YSI6eyJmb3JtYXQiOiJqcGciLCJyZXNpemVfdG9fbGltaXQiOls3MjAsbnVsbF19LCJwdXIiOiJ2YXJpYXRpb24ifX0=--c9426325207613fdd890ee7713353fad711030c7/9673_309.jpg)
An example is provided to illustrate how to compute a composition matrix.
- Two matrices M1 and M2 are taken and the composition matrix is computed by applying M1 followed by M2.
- The process is explained numerically using the matrix entries without relying on visual animations.
- The composition matrix's columns are found by applying M1 to i-hat and j-hat, followed by M2, and using the results.
![An example is provided to illustrate how to compute a composition matrix.](https://www.videogist.co/rails/active_storage/representations/redirect/eyJfcmFpbHMiOnsiZGF0YSI6MTY2Njg5LCJwdXIiOiJibG9iX2lkIn19--85d1ff435f979cc3a68dd1135029a91e552cce38/eyJfcmFpbHMiOnsiZGF0YSI6eyJmb3JtYXQiOiJqcGciLCJyZXNpemVfdG9fbGltaXQiOls3MjAsbnVsbF19LCJwdXIiOiJ2YXJpYXRpb24ifX0=--c9426325207613fdd890ee7713353fad711030c7/9673_309.jpg)
Chapter 5
![A symbolic approach to matrix multiplication is detailed.](https://www.videogist.co/rails/active_storage/representations/redirect/eyJfcmFpbHMiOnsiZGF0YSI6MTY2NjkxLCJwdXIiOiJibG9iX2lkIn19--45179fc443341c366cfefaa799ff9f7b4b0df226/eyJfcmFpbHMiOnsiZGF0YSI6eyJmb3JtYXQiOiJqcGciLCJyZXNpemVfdG9fbGltaXQiOls3MjAsbnVsbF19LCJwdXIiOiJ2YXJpYXRpb24ifX0=--c9426325207613fdd890ee7713353fad711030c7/9673_396.jpg)
A symbolic approach to matrix multiplication is detailed.
- The process for matrix multiplication is generalized using variable entries for any matrices.
- The first column of a composition matrix is determined by multiplying the left matrix by the first column of the right matrix.
- The second column of the composition matrix is determined similarly by multiplying the left matrix by the right matrix's second column.
![A symbolic approach to matrix multiplication is detailed.](https://www.videogist.co/rails/active_storage/representations/redirect/eyJfcmFpbHMiOnsiZGF0YSI6MTY2NjkxLCJwdXIiOiJibG9iX2lkIn19--45179fc443341c366cfefaa799ff9f7b4b0df226/eyJfcmFpbHMiOnsiZGF0YSI6eyJmb3JtYXQiOiJqcGciLCJyZXNpemVfdG9fbGltaXQiOls3MjAsbnVsbF19LCJwdXIiOiJ2YXJpYXRpb24ifX0=--c9426325207613fdd890ee7713353fad711030c7/9673_396.jpg)
Chapter 6
![Associativity of matrix multiplication is explained using the concept of transformation sequencing.](https://www.videogist.co/rails/active_storage/representations/redirect/eyJfcmFpbHMiOnsiZGF0YSI6MTY2NjkzLCJwdXIiOiJibG9iX2lkIn19--3d49a6815b58c9f77803fecc407557e4ea98eb61/eyJfcmFpbHMiOnsiZGF0YSI6eyJmb3JtYXQiOiJqcGciLCJyZXNpemVfdG9fbGltaXQiOls3MjAsbnVsbF19LCJwdXIiOiJ2YXJpYXRpb24ifX0=--c9426325207613fdd890ee7713353fad711030c7/9673_538.jpg)
Associativity of matrix multiplication is explained using the concept of transformation sequencing.
- Associativity means the result of multiplying multiple matrices does not depend on the parenthetical grouping of the matrices.
- The geometric view of matrix multiplication shows that the sequence of transformations is the same regardless of the parenthetical grouping.
- Thinking of matrix multiplication in terms of transformations simplifies the concept and provides a better understanding.
![Associativity of matrix multiplication is explained using the concept of transformation sequencing.](https://www.videogist.co/rails/active_storage/representations/redirect/eyJfcmFpbHMiOnsiZGF0YSI6MTY2NjkzLCJwdXIiOiJibG9iX2lkIn19--3d49a6815b58c9f77803fecc407557e4ea98eb61/eyJfcmFpbHMiOnsiZGF0YSI6eyJmb3JtYXQiOiJqcGciLCJyZXNpemVfdG9fbGltaXQiOls3MjAsbnVsbF19LCJwdXIiOiJ2YXJpYXRpb24ifX0=--c9426325207613fdd890ee7713353fad711030c7/9673_538.jpg)
Chapter 7
![The video concludes with encouragement to practice the concepts and a teaser for the next video.](https://www.videogist.co/rails/active_storage/representations/redirect/eyJfcmFpbHMiOnsiZGF0YSI6MTY2Njk1LCJwdXIiOiJibG9iX2lkIn19--fcae36ff8e196fdcfd84becd71c87cc7252db255/eyJfcmFpbHMiOnsiZGF0YSI6eyJmb3JtYXQiOiJqcGciLCJyZXNpemVfdG9fbGltaXQiOls3MjAsbnVsbF19LCJwdXIiOiJ2YXJpYXRpb24ifX0=--c9426325207613fdd890ee7713353fad711030c7/9673_588.jpg)
The video concludes with encouragement to practice the concepts and a teaser for the next video.
- The viewer is encouraged to play with the concepts and imagine the effects of transformations to solidify understanding.
- A teaser for the next video hints at extending these concepts beyond two dimensions.
![The video concludes with encouragement to practice the concepts and a teaser for the next video.](https://www.videogist.co/rails/active_storage/representations/redirect/eyJfcmFpbHMiOnsiZGF0YSI6MTY2Njk1LCJwdXIiOiJibG9iX2lkIn19--fcae36ff8e196fdcfd84becd71c87cc7252db255/eyJfcmFpbHMiOnsiZGF0YSI6eyJmb3JtYXQiOiJqcGciLCJyZXNpemVfdG9fbGltaXQiOls3MjAsbnVsbF19LCJwdXIiOiJ2YXJpYXRpb24ifX0=--c9426325207613fdd890ee7713353fad711030c7/9673_588.jpg)
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