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Linear Algebra for ML, Probability for ML, Calculus for ML, Chapter 1 …

- - - - ak = c; bk = d
        c/a = d/b = k
        ad = bc
        ad - bc = n
        
        Determinant of
        2x2 matrix = ad - bc
- - - - Zero rows at the bottom
      - Each row as pivot (=Leftmost non-zero entry)
      - Each pivot is to the right of the ones on the rows above
    - - In Row Echelon form
      - Each pivot = 1
      - Any number above pivot = 0
      - Rank = Number of pivots
  - - - Rank = number of rows in matrix – dimension of solution space
      - Solution space of matrix = set of solutions to the equations when the constants are zero
      - Full rank = matrix rank == number of rows
        
        Non-singular only if a matrix has Full rank
        = Row echelon with only 1s in diagonal
      - Rank = largest number of linearly independent rows/columns in the matrix
    - - Pixelated image = matrix ; rank of matrix = amount of space needed to store image
        
        Singular value decomposition (SVD)
        can reduce the rank of matrix while changing it as little as possible
- - - - Orthogonal vectors have dot product 0 <=> 2 vectors are orthogonal if and only if their dot product = 0
      - Geometric dot product
        
        Derivation
      - Equations as dot product
        
        unknowns = column vector
        
        e.g
        
        a+b+c = 10;
        a+2b+c = 15;
        a+b+2c = 12;
- - - - Area of Transformed Fundamental basis = Determinant of transformation matrix
    - - If any matrix in the multiplication is singular the resulting product matrix is singular
  - - - Any point in the system (space) can be represented as a linear combination of vectors in the basis
    - - Points that can be reached by walking in the direction of the vectors in any combination
      - Span of 2 vectors which are not bases = line
        Span of 1 vector = line
      - Basis = Minimal Spanning Set
    - - Basis that sends the original system to another (stretched/shrank) system in original span :question:
    - - Av = λv
        A = transformation matrix,
        v = eigenvector,
        λ = eigenvalue
        
        (A - λI) v = 0
        Since v = non-zero vector,
        (A - λI) = 0 => det(A - λI) = 0
- - - - Skewness: +ive = Left skewed
        -ive = Right skewed
        0 = symmetric around midpoint
      - Kurtosis: Small = Thin tails
        Large = Thick tails
  - - - 68 - 95 - 99.7 Rule
    - - Probability of having x = x1, x2, ... is the same
  - - - distribution of 2 variables
      - Probability of x=x1 and y=y1
        from all possible pairs of x and y
    - - distribution of one variable while ignoring the other
      - reduces the dimension of Joint distribution
        if joint = 3D, marginal = 2D;
- - - - matrix full of second derivatives
      - used to optimize function with multiple variables
      - Determine (local) minima | maxima
        
        Calculate Hessian matrix of a certain point
        
        Calculate λ values from det(H(x,y) - λI) = 0
        
        If all vales of λ > 0 => minimum;
        If all values of λ < 0 => maximum;
        If λ > 0 & λ < 0 => saddle point => can't determine
      - Example of using Newton's method
        for optimization of multiple variables