Abstract vector spaces

Back to the beginning, what is vector?
Description
Function is vector?
- Yes!
- It is another type of vector with different basis.
  - example : In polynomial space, we can express
  \[x^2 + 3x + 5 = \begin{bmatrix} 5\\3\\1 \end{bmatrix},b_0(x) = 1,b_1(x)=x,b_2(x) = x^2\]
- We can say all of things as vector that satisfy axiom of vector!
  - Rule for vectors addition and scaling
    - \[\vec{u} + (\vec{v}+\vec{w}) = (\vec{u} + \vec{v})+\vec{w}\]
    - \[\vec{v}+\vec{w} = \vec{w}+\vec{v}\]
    - \[\vec{0} + \vec{v} = \vec{v}\]
    - \[- \vec{v} + \vec{v} = \vec{0}\]
    - \[a(b\vec{v}) = (ab)\vec{v}\]
    - \[1\vec{v} = \vec{v}\]
    - \[a(\vec{v}+\vec{w} ) = a\vec{v}+a\vec{w}\]
    - \[(a+b)\vec{v} = a\vec{v} + b\vec{v}\]
  - In the modern theory the form that vectors take doesn’t really matter.
    Next Step
review! and solve a mount of problems!
References
https://www.youtube.com/watch?v=PFDu9oVAE-g&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=15

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