A-Z Maths for Data Science.

Understand that the transpose swaps rows and columns, reversing the shape (2x3 to 3x2; 3x3 example), and note the matrix-transpose relationship to explore after learning basic operations.

Compute a vector’s magnitude using the L2 norm or Euclidean distance, and convert any vector to a unit vector by dividing by its magnitude, with 2D and 3D examples.

Multiply a vector by a scalar by applying the scalar to each element, scaling the magnitude while preserving direction; a negative scalar reverses direction.

Explore the distributive properties of vectors and scalars, and define the angle between two vectors as theta, with examples on the x and y axes, often using the smallest angle.

Compute the angle between two vectors using the dot product and magnitudes, illustrated with v1=(1,0) and v2=(0,1); cos theta = (v1 dot v2)/(|v1||v2|), theta=90 degrees.

Master orthonormal vectors. Divide by the magnitude to obtain a unit vector from orthogonal vectors with unit magnitude, and use the dot product to relate magnitudes to cosine of angle.

Explore how a line is represented in vector form as w^T x + w0 = 0, and how w0 indicates whether the line passes through the origin.

Learn to project a vector onto a line, using coordinates or angles, with a projection equal to the vector magnitude times the cosine of the angle between them.

Learn how line equations use W^T x and W^T x + w0 to define points via signed distance, and how these ideas extend to circles, spheres, and higher dimensions.

Explore matrices as 2d arrays and distinguish vectors as 1d arrays, explain rows, columns, and matrix size m x n, with element notation A11.

Master matrix operations: element-wise addition and subtraction, Hadamard product, and dot-product multiplication with shape compatibility. Apply these ideas to linear equations and matrix representations.

Learn how an orthonormal matrix has every row and column as an orthonormal vector, so A^T A equals the identity; examine identity, diagonal, and symmetric matrices as orthonormal examples.

Explore how to compute determinants using minors and cofactors, and derive a matrix inverse by applying these concepts to 2x2 and 3x3 cases.

Discover how to compute the inverse of a square matrix using determinant, cofactors, adjoint, and transpose, yielding the identity matrix; learn the orthogonal matrix property where transpose equals inverse.

Learn how the probability mass function represents discrete distributions and how the probability distribution function applies to continuous cases, illustrated with a fair dice example.

Explore continuous random variables and the probability density function, height distributions, that the area under the curve equals one and probability between two values; exact heights have zero probability.

Explore the Bernoulli distribution, a discrete two-outcome model where a success occurs with probability p and a failure with one minus p, with examples like heads or tails.

Explore the law of large numbers and how the average of many trials converges to the expected value, illustrated with dice rolls.

Explore the normal distribution by examining its formula, parameters mu and standard deviation, and how changing these values reshapes the curve for each x using real data in Excel.

Understand the z score, defined as (x - mu)/sigma, and how many standard deviations a value lies from the mean in both original and unit normal distributions.

Explore symmetric distributions and skewness, show how in a normal distribution the mean equals the mode equals the median, and describe how positive or negative skew affects tails and moments.

Central limit theorem states that sample means are normally distributed with mean mu and standard deviation sigma/√n; for population, use n≥30, but if the population is normal, any n works.

Apply the central limit theorem to find the probability that the sample mean of 49 shoppers lies between 441 and 446, with mu 448 and sigma 21.

Explore discrete and continuous uniform distributions, with dice illustrating equal likelihood, compare to the normal distribution, and relate area under the curve to probability.

Test the null mu=20 vs alt mu<20 with n=20, mean 19.8, s=3.1 using a t-score; interpret the p-value and fail to reject the null.

Conduct a one-tailed t-test comparing mu=82 to mu>82 for a 25-student sample; with t=3.65 and alpha=0.05, reject the null and support the alternative.

Explore mutually exclusive and independent events, understand how they affect probability, and apply formulas for union, intersection, and conditional probability with clear examples.

Understand probability by prioritizing question comprehension and using formulas as support when helpful. Explore examples, circular or odd plates and garment shop or second floor, highlighting intuition and union-intersection methods.

explains conditional probability and chain rule using a 52-card deal to four players, calculating the probability that each gets one ace.

Explore the birthday paradox and why intuition misleads. Compute the probability with a 365-day year: denominator 365^10 and numerator 365×364×...×356, yielding no two share a birthday about 0.8803.

Explore probability through cricket scenarios and reliability problems, applying intersection and conditional probability, and using independence to compute the chance that all ten components work: (0.99)^10.

Visualize complex probability problems with probability trees, map events, and calculate intersections like P(A ∩ B) using diagrammatic decision trees for ad-skipping scenarios.

Explore the total law of probability, partitioning the sample space into mutually exclusive events, and compute P(A) from P(A∩B) and P(A∩C) using conditional probabilities.

Use the total probability theorem and conditional probability to determine if a tape recorder that died within six months had a flaw, based on flaw and death rates.

Explore the total law of probability with real-world examples, from the elevator scenario to five independent questions using binomial reasoning (at least two, not all five).

Explore how counting underpins probability, distinguishing permutations and combinations, where order matters or not, and learn to apply formulas to complex selection problems.

Explore how constraints and dependencies alter the counting of travel combinations across two legs, illustrating mutual exclusivity, independence, and conditional probability in permutations and combinations.

Learn to decide when order matters or not by solving a one-from-each-discipline selection and counting five-digit ternary sequences, addressing independence and first-digit constraints.

Discover factorial notation, where n! is the product of 1 to n, and n! = n × (n−1)! to simplify calculations.

Explore permutation as an ordered arrangement of k distinct elements from n, derive the formula nPk = n!/(n-k)!, and verify with examples like 5P2.

Explore permutation with repetition using the formula, counting arrangements of probability and Mississippi when letters repeat, and apply factorials to handle repeated letters.

Explore counting permutations when vowels appear together by treating them as a single unit with consonants, and expand the vowel group to account for internal arrangements and repeats.

Explore permutation methods by arranging letters in correction with vowels together as a unit, and in collection with vowels not together, accounting for repeats and factorial terms.

Learn permutations with repetition via a two-digit example using A, B, C to yield 9 ways, and observe the 3^n growth when N and K repeat.

Learn how combinations differ from permutations by exploring unordered selections of k distinct elements from n, using nCk formulas and practical examples like committees.

Compute the expected value of an action by weighting outcomes by their probabilities, and see why repeated trials reveal the law of large numbers.

Explore the expected value of A X plus minus B. Note E[X+Y] = E[X] + E[Y], and Var(X+Y) = Var(X) + Var(Y) only if X and Y are independent.

Compute the expected total weight using X and Y and the rule E[X+Y]=E[X]+E[Y]. Evaluate independence by comparing P(Y|X) to P(Y) in the sweet and mixture example.

Draw two balls from a bag of eight white, three black, and two red; black yields ten rupees, white minus two, red zero; compute expected value and probabilities for profitability.