Matrices and Linear Transformations

from University of Tokyo 1S1 Mathematical Sciences Foundation: Linear Algebra Matrices and Linear Transformations

• In short,
  • The transformation mappings of vectors that can be represented by matrices have Linearity.
  • This is because the Distributive Law and the Associative Law of Scalar Multiplication hold in Matrices and Their Operations.
• What is a Mapping defined by a matrix?
  • The mapping $F:R^n\to R^m$ can be represented by an m x n matrix.
  • 
  • The transformation of vectors corresponds to multiplication of vectors and matrices.
  • It is a generalization of Linear transformations on a plane, so the same discussion applies (such as composition of mappings).
• What is a Linear Mapping?
  • A Mapping that satisfies Linearity.
  • It can be considered as a generalization of Linear Functions and Linear Transformations.

Linear Functions	Linear transformations on a plane (Linear Transformations)	Linear Mapping
Can be represented by real numbers (1x1 matrices)	Can be represented by 2x2 matrices	Can be represented by m x n matrices
Transforms real numbers (1-dimensional vectors)	Transforms 2-dimensional vectors	Transforms n-dimensional vectors

    - I don't think it's harmful to not worry about the differences between functions, mappings, and transformations (takker)(nishio)
        - There are only minor differences in notation.
        - Some authors may use them differently.
            - For example, in "[まずはこの一冊から 意味がわかる線形代数 (BERET SCIENCE)]", "transformation" is defined as "a mapping where the dimensions of the input and output are equal".
            - In such cases, you should follow that definition only for that discussion.
        - [Mapping - Wikipedia](https://ja.wikipedia.org/wiki/写像)
        - > Functions, transformations, operators, and arrows are sometimes used as synonyms for mappings.
        - I see (blu3mo)
            - I mistakenly thought that functions can only accept real numbers (1x1 matrices), but that's not the case.
            - Depending on the person, it may be that they are giving such a definition to functions, so it cannot be simply called a misunderstanding (takker)
                - As I mentioned before, I think it's a matter of different notations, and if the textbook or instructor has given their own definition, it's best to follow it.
- Does that make sense? (blu3mo)
    - I also want to consider the geometric image (blu3mo)
- Geometric Image
    - For example, a mapping that rotates a vector by θ is a linear mapping.
        - Because,
            - Rotating two vectors and then adding them is the same as adding them and then rotating.
            - Scaling a vector and then rotating it is the same as rotating it and then scaling it.
- On the other hand, what is a non-linear mapping? (blu3mo)
    - Since it can be represented by a matrix, it should be a linear mapping with the properties of the distributive law, etc.
        - There are many mappings that cannot be represented by matrices (nishio)
            - For example, $f(x)=x^2$ cannot be represented by a matrix.
            - Oh, it's written just below, haha.
    - So, should we consider mappings that cannot be represented by matrices?
        - There can be as many as you want.
- Does a linear mapping need to have a [[Square Matrix]] as its representation vector? (blu3mo)
    - [[Linear transformations on a plane]] were square matrices.
    - Ah, but it's just a matter of not getting the same dimension output as the input if it's not a square matrix.
        - If you want to change an n-dimensional vector to another n-dimensional vector, it must be a [[Square Matrix]].
        - For example, in the case of plane vectors to plane vectors, it must be a 2x2 matrix.