Basic Mathematics of Data Science

Mathematics as a discipline

Mathematics, as a science, is one of the oldest known forms of logical thinking by mankind. Since ancient Mesopotamia and likely before (3,000 BCE), humans have been relying on arithmetic and more challenging forms of math to answer life’s biggest questions. Basic Mathematics Today, we rely on math for most aspects of our daily life.

Basic symbols and terminology

First, let’s take a look at the most basic symbols that are used in the mathematical process as well as some more subtle notations used by data scientists.

Vectors and matrices

Logarithms/exponents

Key Points

  • An exponent of −1−1 denotes the inverse function. That is, f−1(x)f−1(x) is the inverse of the function f(x)f(x).
  • An inverse function is a function that undoes another function: If an input xx into the function ff produces an output yy, then inputting yy into the inverse function gg produces the output xx, and vice versa (i.e., f(x)=yf(x)=y, and g(y)=xg(y)=x).
  • The logarithm to base bb is the inverse function of f(x)=bxf(x)=bx: logb(b)x=xlogb(b)=xlogb⁡(b)x=xlogb⁡(b)=x
  • The natural logarithm ln(x)ln(x) is the inverse of the exponential function exex:b=elnbb=elnb

Set theory

Calculus

  • Functions of a single variable, limit, continuity, differentiability
  • Mean value theorems, indeterminate forms, L’Hospital’s rule
  • Maxima and minima
  • Product and chain rule
  • Taylor’s series, infinite series summation/integration concepts
  • Fundamental and mean value-theorems of integral calculus, evaluation of definite and improper integrals
  • Beta and gamma functions
  • Functions of multiple variables, limit, continuity, partial derivatives
  • Basics of ordinary and partial differential equations

Linear Algebra

  • Basic properties of matrix and vectors: scalar multiplication, linear transformation, transpose, conjugate, rank, determinant
  • Inner and outer products, matrix multiplication rule and various algorithms, matrix inverse
  • Special matrices: square matrix, identity matrix, triangular matrix, an idea about sparse and dense matrix, unit vectors, symmetric matrix, Hermitian, skew-Hermitian and unitary matrices
  • Matrix factorization concept/LU decomposition, Gaussian/Gauss-Jordan elimination, solving Ax=b linear system of equation
  • Vector space, basis, span, orthogonality, orthonormality, linear least square
  • Eigenvalues, eigenvectors, diagonalization, singular value decomposition
  • Sinan Ozdemir-Principles of Data Science (Packt)

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Desi Ratna Ningsih

Desi Ratna Ningsih

Data Science Enthusiast, Remote Worker, Course Trainer, Archery Coach, Psychology and Philosophy Student