Mathematics-Basics to Advanced for Data Science And GenAI

Explore core linear algebra concepts: vectors, matrices, linear transformations, eigenvalues, and PCA, and their use in data representation, regression, neural networks, and dimensionality reduction for data science.

Scalar is a single numerical value with magnitude; a vector is an ordered list of numbers with magnitude and direction to represent data points in space for data science tasks.

Explore adding vectors across dimensions, from 2D to multi-dimensional space, and apply vector addition to data science tasks like data aggregation, feature engineering, NLP word embeddings, and image processing.

Understand the multiplication of vectors, including dot product (inner product), element-wise and scalar multiplication, with applications in data science and nlp, plus cosine similarity and Euclidean and Manhattan norms.

Explore vector databases and cosine similarity to enable a retrieval augmented generation (rag) system that answers questions from documents by converting text to vectors and querying a vector store.

Learn element wise multiplication, multiplying corresponding elements to form a new vector of the same dimension, with data science applications like feature engineering and neural networks.

Master scalar multiplication of vectors, where a scalar scales every component (e.g., (3,5,7)×4). Also cover dot/inner/elementwise products, normalization and standardization, including 0–255 to 0–1 scaling and cm-to-m conversion.

Explore matrices as rectangular arrays with rows and columns, enabling data representation, image and text handling, and applications in neural networks, regression, and NLP.

Explore fundamental matrix operations—addition and subtraction of same-dimension matrices, scalar multiplication, and matrix multiplication via dot products. See how these tools analyze multi-dimensional data in data science.

Explore functions and linear transformations in linear algebra, and see how inputs map to outputs, underpinning dimensionality reduction and machine learning, with notations like f: X -> Y.

Explore vector transformations that map vectors from one space to another, changing magnitude and direction, with examples like scaling, rotation, and reflection, and applications in data science and computer graphics.

Explore linear transformations as functions preserving vector addition and scalar multiplication, illustrated by a reflection across the y-axis and a fixed-vector translation that fails linearity.

Discover why linear transformation underpins data science, enabling dimensionality reduction with principal components, feature engineering, normalization, standardization, and applications in regression and neural networks.

Explore how linear transformations preserve the origin, map lines to lines, and satisfy homogeneity and additivity, illustrated with t(x)=2x and 90-degree rotation.

Compute the magnitude (length) of vectors using the Pythagorean formula for any dimension. Derive unit vectors by normalizing vectors with their magnitude, and apply normalization in data scaling.

Explore projections in linear algebra by projecting a vector onto a line using the dot product and the formula x·v / v·v times v, noting the projection is perpendicular to the line.

Explore the inverse of a function, its existence and uniqueness, and how inverse mappings restore original inputs, highlighting injective and bijective conditions and data science applications.

Explore how inverse functions enable normalization and standardization in data science, applying z-score standardization, min-max scaling, and inverse transforms to interpret model predictions.

Compute the inverse of a 2x2 matrix using its determinant and the standard formula. Verify the result by applying a vector transformation and recovering the original vector.

Explore eigenvalues and eigenvectors through matrix transformations, determinants, and visualization, showing how eigenvectors retain direction while scaling, with applications to principal component analysis in data science.

Explore equations of lines, planes, and hyperplanes across dimensions, such as y=mx+c and w^T x + b = 0, highlighting slope, intercept, and origin cases.

Understand statistics as the science of collecting, organizing, and analyzing data and its use in data analytics for eda, feature engineering, feature selection, and decision making.

Explore descriptive and inferential statistics, including central tendency, dispersion, data distribution, and five-number summaries. Learn hypothesis testing and confidence intervals with Python.

Define population data as the complete set and sample data as a representative subset, then explain parameters vs statistics with examples like mu, sigma squared, and practical uses.

Delve into probability and non-probability sampling, covering simple random, systematic, stratified, cluster, multi-stage, convenience, judgmental, snowball, and quota methods, with exit polls in a data science context.

Classify data types into quantitative and qualitative, with discrete and continuous, and nominal and ordinal, using examples from bank accounts to temperatures to guide data analysis and EDA.

explore scales of measurement for data, including nominal, ordinal, interval, and ratio, and learn how categories, ordering, differences, and true zero influence analysis.

Define and compare mean, median, and mode, including population mean and sample mean, and show how distribution and outliers guide their use in feature engineering.

Explore the measure of dispersion, defining range, variance, standard deviation, and interquartile range; learn how range is max minus min and how variance and standard deviation describe data spread.

Explore why sample variance uses n minus one in the denominator instead of n. See how basal correction and degree of freedom enable unbiased estimation with x-bar, mu, and sigma-square.

Learn how random variables—functions from experiments like coin tosses and dice rolls—take discrete values or continuous ranges. Explore probability density and distribution concepts.

Krish defines percentile as the value below which a percentage of observations lie, using the counts below x over n times 100, and introduces quartiles (25th, 50th, 75th) with averaging.

Learn the five-number summary—minimum, q1, median, q3, maximum—and use iqr and fences to detect outliers, visualize with a box plot, and apply to feature selection in data science.

Explore how histograms visualize data distributions, frequencies, and binning, and how skewness reveals symmetry, with examples of symmetric, right-skewed and left-skewed data, box plots, and mean, median, and mode relationships.

Learn covariance and correlation, including positive and negative relationships between two continuous variables. Compare Pearson and Spearman coefficients and apply them to feature selection in data science.

Explore basic probability concepts, including the additive rule for mutually exclusive and non-mutually exclusive events, with coin tosses, dice, and cards, and the multiplication rule for independent and dependent events.

Explore the multiplication rule for independent and dependent events with coin tosses, dice rolls, and card draws. Recognize conditional probability and Bayes theorem in machine learning.

Explore probability distribution functions and types, including PMF for discrete variables, PDF for continuous variables, and CDF as a cumulative density function, with histograms, density, and area under the curve.

Explore the types of probability distributions, from Bernoulli and binomial to normal, Poisson, log-normal, and Gaussian. Learn how pdf, pmf, and cdf relate to discrete and continuous data.

Explore the Bernoulli distribution, a discrete binary model with pmf, two outcomes 0 and 1, and learn to compute mean, variance, and standard deviation with real-world examples.

Explore binomial distribution, a sequence of n independent Bernoulli trials with binary outcomes and its pmf, mean np, and variance npq. See how coin tosses illustrate exactly k successes.

Explore the Poisson distribution, a discrete probability distribution, and learn to construct its PMF using lambda as the expected number of events in a fixed time interval, and compute probabilities.

Explore the Gaussian distribution, a symmetric bell curve with mean equal to the median and the mode, illustrated by iris data set, weights, and heights, plus the empirical rule 68-95-99.7.

Explore the standard normal distribution and z scores to transform data to mean zero and unit variance, enabling standardization for machine learning models and exploratory data analysis.

Understand the uniform distribution in both continuous and discrete forms, including pdf and cdf for continuous bounds a and b, and pmf for discrete outcomes like dice.

Explore the log normal distribution, its connection to the normal via natural log, its right-skewness, and a Q-Q plot, with examples like wealth, comment length, chess games, and dwell time.

Explore power law distribution, its 80/20 rule and long tail, with examples from IPL wins, wealth, oil, and word frequencies; learn Box-cox transforms and q-q plots for normality.

Explore the pareto distribution as a non-Gaussian, power-law form of the 80/20 rule, emphasizing alpha in the pdf and how Box-Cox transformation converts Pareto data to a normal distribution.

Explore how the central limit theorem uses sampling distributions to show that the mean becomes normally distributed as sample size grows, regardless of population shape.

Explore inferential statistics by defining estimates, distinguishing point estimates like the sample mean from interval estimates and confidence intervals for unknown population parameters.