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calculus-and-linear-algebra.md

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Linear Algebra

Concepts

  • Vectors and Matrix
    • vectors
    • matrix operations
      • invertible and singular matrix
      • relationship between linear system and matrix
      • permutation matrix
      • Gaussian elimination
        • Echelon matrix
        • pivot variables, (# = C(A)), free variable(# = N(A))
        • special solutions
        • Null matrix
      • matrix factorization
  • Vector Space
    • definition
      • commutative, associative, distributive, unique identity
      • uniqueness of identity and inverse
    • row space, column space C(A)
    • null space N(A)
    • Linearly dependent
    • rank
    • span
      • rank, span and solution of linear system
    • basis
      • basis of matrix spaces and function spaces
    • dimension

Questions

  • Solve a linear equation
  • relationship between Ax=b and Ax = 0
    • special solution, rank, and linear space
  • when is a matrix invertible (A-1b as only solution)
  • full row rank vs full column rank
    • full column rank
      • all columns are pivot columns
      • no free variables (null space 0 dimension)
      • invertible, determined
    • full row rank
      • columns spans whole space
      • n - r = n - m solutions
      • if m < n, undetermined

Calculus

  1. $$(lnx^{lnx})'$$
    1. use the log trick and chain rule
  2. Integrate
    1. ln(x)
    2. sec(x)
      1. method1 : multiplies $$\frac{secx+tanx}{secx+tanx}$$
      2. method2: sec(x) = 1/cosx = cosx/cos^2 x = cosx / (1- sin^2x )
    3. E(X|X>0) for normal random variable
    4. total area under curve for normal p.d.f
      1. Calculate $$\int_{-\infty}^{\infty} e^{-x^2/2} \int_{-\infty}^{\infty} e^{-y^2/2}$$ and use polar (rcos, rsin) axis (Fubini's Rule)
  3. Calculate volume/distance/mass/probability using integration
    1. snow begins 12pm, snow plow can clear constant volume per minute, 1pm moved 2miles, 2pm, 3 miles, when did the snow fall?
  4. Proof
    1. Mean-Value Theorem
    2. Newton's Method
  5. Concepts
    1. Remann vs Lebesgue
  6. Tricks
    1. x^x^... = 2, x = ? if = 4, x = ?

Algebra

  • 100th digit of (1+sqrt(2))^3000
    • use Binomial Theorem

Stochastic Calculus

  • Brownian motion/Wiener process
    • zero mean
    • normal i.i.d. increments with variance dt
    • continuous
    • Markov
  • First passage time
    • Martingale stops at stopping time
    • reflection principle
  • Martingale
    • B(t)-t, exponential martingale
  • Ito's Lemma