Mapping

from University of Tokyo 1S1 Mathematical Sciences Foundation: Linear Algebra Mapping

• What is a mapping?
• Generalization of functions
  • Seems interesting (blu3mo)
  • It’s like making the domain more general (blu3mo)
  • Honestly, it’s not wrong to say that a mapping is a function with the domain and range restricted to numbers (takker)
    • That’s the extent of it
• For example, $f(x)=x^2$ is a mapping from $R$ to $R$
• Definition
  • Given sets X and Y,
  • A mapping $f:X\to Y$ from X to Y is
    • A rule that assigns a unique element $f(x)$ of Y to each element x of X
      • The “unique” part is important (blu3mo)
        • We learned this as the definition of a function in IB (blu3mo)
      • If it’s not unique, then the notation $f(x)$ doesn’t even make sense (takker)
• Examples
  • If we write $f(x)=x^2$ as a mapping, it would be written as
    • $$\begin{gathered} f:R\to R \\ x\mapsto x^2 \end{gathered}$$
    • Write the sets of inputs and outputs, then define the \mapsto
  • Even things that don’t look like traditional functions are mappings
    • Addition is also a mapping
      • $$\begin{gathered} h:R^2\to R \\ (x,y)\mapsto x+y \end{gathered}$$
      • This can be written as a mapping from ℝ^2 (2-dimensional) to ℝ (1-dimensional)
    • I see, this is similar to how addition and other operations can be treated as functions in Haskell and other languages that use infix notation (blu3mo)
      • I thought so too, but it’s actually different. The symbol for addition is not defined separately. It’s just wrapping it in a function. Well, addition is a mapping from $R^2$ to ℝ (takker)
    • I wrote about variations in notation like this in the chat (takker)
      • I’ll link it later
  • The identity mapping is a mapping where the inputs and outputs are the same
    • If we write the definition, it would be
    • $$\begin{gathered} i{d}_X:X\to X \\ x\mapsto x \end{gathered}$$
      • The set and the elements are the same
        • If the sets were different, even if the correspondences of the elements were the same, it wouldn’t be the identity mapping (blu3mo)
          • It’s important to think about the definition
      • Here, X represents a set, so it should be $i{d}_X$, not $i{d}_x$
• Injection
  • In $X\to Y$,
  • Every x corresponds to a different y
  • This depends on what X is (Y doesn’t matter) (blu3mo)
• Surjection
  • In $X\to Y$,
  • There exists an x that corresponds to every y
    • This depends on what Y is (X doesn’t matter) (blu3mo)
      • $$\begin{gathered} h:R^2\to R \\ (x,y)\mapsto x+y \end{gathered}$$ is not surjective
      • $$\begin{gathered} h:R^2\to R^{+} \\ (x,y)\mapsto x+y \end{gathered}$$ is surjective
  • As a premise, there exists a y corresponding to every x, so
    • It means that every x and y are paired
    • (Duplicates are possible, so it’s not a bijection)
• Bijection
  • Both an injection and a surjection
  • In other words, there is only one y corresponding to each x
• 
  • All injections are injections
    • Are there no x such that f(x)≦0? (blu3mo)
    • If it’s a bijection, there must be an inverse mapping, so you can judge from there too (takker)
    • f(x) is 0 for x=0,1, so it’s not an injection
• In visual terms,
  • Injection:
  • Surjection: There is a point on the graph at every height (y)
• Composition of mappings
  • $g\circ f(x):=g(f(x))$ is basically the same as a function
  • The important thing is that $f:X\to Y,g:Y\to Z$
    • Basically, if the output of f(x) and the input of g don’t match, you can’t compose them
    • We haven’t been conscious of this because it was mostly $R^1$, but it’s important
    • It’s like static typing in programming (blu3mo)
      • Math = programming (somewhat extreme argument) (takker)
      • This realization is important (takker)
• Inverse mapping
  • Just like with composition of mappings, it’s important to be aware of whether the types match
  • Definition: If $g\circ f=i{d}_X$ and $f\circ g=i{d}_X$, then $f^{-1}:=g$
    • I see~
    • As an example where it doesn’t work if only one of them is true,
      • 
• Theorem: Inverse mapping exists if it’s a bijection- Can’t we say that if this is surjective, then the inverse mapping exists?
  • No, if it is not injective, then considering the inverse mapping, there won’t be a unique x for y, which violates 624e6a9979e1130000baa336.
• I feel like I can see the relationship between surjectivity and injectivity (blu3mo).