Abstract vector spaces

What to learn

  • 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