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Mathematics in Machine Learning, one map just for linear algebra soon, and…

- - - - Point-slope Form
      - Standard Form
      - Slope-intercept Form
  - - - Forward Elimination
      - pivots
        
        pivots can never be 0 tn. don't accept 0 as a pivot
        
        if the pivot is 0, then its a failure, if there a more rows available, try to exchange rows, to prevent getting 0
        
        [[1,2,0][0,2,3][004]]
        
        1, 2, 4 = pivots
  - - - Column Spaces
      - Null spaces
        
        Ax = 0
      - "Have to go through the zero origin (0,0,0) = for example in a 3-Dimensional plane that's why it have (x,y,z) nice explanation
        
        if it does not go through the origin it is not a subspace
      - Row Space ? need to check if its a part of subspaces
        
        C(A^t)
      - Left Null Row Space ? need to check if its a part of subspaces
        
        N(A^t)
    - - independent
        
        if no combinations gives [zero vector], except the zero combination, otherwise its dependent
    - - Combination of vectors
    - - question like: what are the basis of the space(? need review)
      - basis
        
        eg: independent, same number of dimensions, they span the space... (infinitely many more(?))
  - - - a graph without loops is called a tree
    - - Tells what the graph tells, in matrix form
  - - - Perpendicuar
      - 90deg right triangle
      - x^t*y = 0
        
        No always true
      - Orthonormal
        
        independent columns
        
        always independent
        
        gram-schmidt
        
        in elimination we find the triangular matrix, in gram-schmidt the goal is to find the orthonormal matrix
        
        A = QR
  - - - find the point on the line that is closest to b
    - - Linear Rregression
        
        as long as the columns of A are Independent
      - not good with outliers
  - - - odd(-), even(+)
    - - Imagine a box, determinants is a good formula for finding the area
    - - Not as efficient as Elimination
  - - - det(A - lambda I) = 0
    - - (A - lambda I)
      - Should be independent? gives the whole story of the matrices
  - - - Q = ⊥
  - - - Eigenvalue test : λ1 > 0, λ2 > 0
      - Determinants Test: a > 0, ac-b^2 > 0
      - Pivot Test: a > 0, ac-b^2/a > 0
      - x^TAx > 0, except x = 0
  - - - B = M^-1AM
  - - - U = orthogonal
      - Σ = diagonal
      - V = orthogonal
      - finding V : A^TA = VΣ^TΣV
        
        eigenvectors = V's
        
        eigenvalues = Σ's
      - finding U: AA^T = UΣΣ^TU^T
  - - - Finding our A
  - - - Relation between A to B
        
        B = M^-1AM
  - - - has one or zero solutions
        
        full column rank
    - - infinitely many solutions
        
        full row raw
    - - the projection matrix
    - - full rank r = m = n