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Linear Algebra - Coggle Diagram

- - - - \((a\cdot b)\odot v=a\odot(b\odot v)\)
    - - \(a\odot(u\oplus v)=a\odot u\oplus a\odot v\)
    - - \((a+b)\odot v=a\odot v\oplus b\odot v\)
- - - - square matrix whose columns of entries are unit vectors perpendicular to each other
    - - determinant is plus-minus one
  - - - has \(n\) eigenvectors which are the standard basis of \(\mathbb{R}^n\)
      - each eigenvector \(\vec{v}_i\) is scaled by a factor of the top-left diagonal entry \(d_i\) (eigenvalue)
      - the determinant is the product of the top-left diagonal entries
  - - - \(n\times n\) matrix whose entries are symmetric over the top-left diagonal
    - - decomposition
        
        there exists a linear map
        
        \(T_D\) which scales each basis vector \(\vec{b}_i\) in \(\mathbb{R}^n\) with the eigenvalue \(\lambda_i\)
        
        \(T_O\) which rotates each
        basis vector \(\vec{b}_i\) in \(\mathbb{R}^n\) to the eigenvector \(\vec{v}_i\)
        
        composition
        
        rotate the eigenvectors to the standard basis of \(\mathbb{R}^n\), scale them, and rotate them back (\(T_S=T_O\circ T_D\circ\mathrm{inv}(T_O)\)).
      - has \(n\) eigenvectors perpendicular to each other
      - each eigenvector \(\vec{v}_i\) is scaled by the eigenvalue \(\lambda_i\)
  - - - decomposition
        
        there exists a linear map
        
        \(T_{A_{m\times n}}\) which maps each vector from \(\mathbb{R}^n\) to \(\mathbb{R}^m\)
        
        \(T_{D_{m\times m}}\) which scales each basis vector \(b_i\) with the square root of the nonzero eigenvalue \(\lambda_i\) of \(T_S\)
        
        \(T_{O_{n\times n}}\) which rotates each
        basis vector \(\vec{b}_i\) in \(\mathbb{R}^n\) to the eigenvector \(\vec{v}_i\) of \(T_{S_{n\times n}}\)
        
        \(T_{O_{n\times n}}\) which rotates each
        basis vector \(\vec{b}_i\) in \(\mathbb{R}^n\) to the eigenvector \(\vec{v}_i\) of \(T_{S_{m\times m}}\)
- - - - \([x]_B=\begin{bmatrix}x_1 \\ \vdots \\ x_n \end{bmatrix}\)
      - Only basis such that coordinate vector of \(x\) has the same elements as \(x\) itself.
    - - n-tuples of real numbers \(\Leftrightarrow x=(x_1,\ldots,x_n)\in F^n\)
- - - - \(f(\phi_A(a, b))=f(a\vec{u}+b\vec{v})=af(\vec{u})+bf(\vec{v})\)
      - \(af(u_1,u_2)+bf(v_1,v_2)=a(u_1+ 2u_2,3u_1+ 4u_2,5u_1 + 6u_2)\)
        \(+b(v_1+ 2v_2,3v_1+ 4v_2,5v_1 + 6v_2)\)
      - Let \(B=\{\vec{u},\vec{v}\}\) be the ordered set of basis vectors to \(\mathbb{R}^2\), where \(\vec{u}=(u_1,u_2)\) and \(\vec{v}=(v_1,v_2)\)
      - \(af(\vec{u})+bf(\vec{v})=af(u_1,u_2)+bf(v_1,v_2)\)
- - - - \(f(k\odot v)=k\circ f(v)\in W\)
      - \(f(u\oplus v)=f(u)\diamond f(v)\in W\)