IIT M FOUNDATION AN EXAM QDF2 26 Oct - 26 Oct 2025

Concepts this paper tests

Auto-generated summary from the source site. Handy as a checklist; the lessons above explain each idea from scratch.

Here’s a structured summary of core topics, concepts, principles, and equations for the "Diploma Level: Machine Learning Foundations" exam, along with Mermaid knowledge graphs to visualize relationships.

1. Core Topics & Concepts

A. Machine Learning Basics

1. Supervised Learning

   • Classification (e.g., loan approval: Yes/No based on features like income, credit score).

     • Algorithms: Logistic Regression, Decision Trees, SVM, k-NN.

   • Regression (predicting continuous values, e.g., house prices).

     • Algorithms: Linear Regression, Polynomial Regression.

   • Loss Functions:

     • Squared Loss (MSE): $L(y, \hat{y}) = (y - \hat{y})^2$ (used in regression).
     • Log Loss (Cross-Entropy): Used in classification.

1. Unsupervised Learning

   • Clustering (e.g., customer segmentation).

     • Algorithms: k-Means, Hierarchical Clustering.
   • Dimensionality Reduction (e.g., PCA, t-SNE).

1. Model Evaluation

   • Overfitting/Underfitting.
   • Train-Test Split, Cross-Validation.

B. Linear Algebra (Critical for ML)

1. Matrices & Vectors

   • Row Space: Span of the rows of a matrix $A$.

   • Basis: Linearly independent vectors that span a space.

     • Example: For $A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & -1 \\ 0 & 0 & -1 \end{bmatrix}$, a basis for the row space is {(1,0,2), (0,1,-1), (0,0,-1)}.
   • Rank: Number of linearly independent rows/columns.

1. Matrix Properties

   • Orthogonal Matrix: $A^T A = I$ (columns are orthonormal).
   • Invertible Matrix: $\det(A) \neq 0$, $AA^{-1} = I$.
   • Symmetric Matrix: $A = A^T$.

   • Eigenvalues/Eigenvectors:

     • $A \mathbf{v} = \lambda \mathbf{v}$.
     • Real Symmetric Matrices: Always have real eigenvalues.
     • Diagonalizable Matrix: $A = PDP^{-1}$ (requires $n$ linearly independent eigenvectors, not necessarily distinct eigenvalues).

1. Special Matrices

   • Positive Definite: All eigenvalues > 0.
   • Singular Matrix: $\det(A) = 0$ (non-invertible).

C. Calculus & Optimization (For ML Models)

1. Gradient Descent:

   • Update rule: $\theta = \theta - \alpha \nabla J(\theta)$.
   • Used to minimize loss functions (e.g., MSE, Log Loss).

1. Partial Derivatives:

   • Critical for computing gradients in multivariate functions.

D. Probability & Statistics (For ML)

1. Probability Distributions:

   • Gaussian (Normal), Bernoulli, Binomial.

2. Bayes’ Theorem:

   • $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$.

3. Maximum Likelihood Estimation (MLE):

   • Finds parameters that maximize the likelihood of observed data.

2. Key Equations

Graph 2: Linear Algebra Concepts

Graph 3: Optimization in ML

Graph 4: Probability & Statistics

4. Exam Tips

1. Classification vs. Regression:

   • Use classification for discrete outputs (e.g., Yes/No).
   • Use regression for continuous outputs (e.g., house price).

1. Linear Algebra:

   • A matrix is invertible iff $\det(A) \neq 0$.
   • A matrix is orthogonal iff $A^T A = I$.
   • Real symmetric matrices always have real eigenvalues.
   • Diagonalizable ≠ n distinct eigenvalues (only needs $n$ linearly independent eigenvectors).

1. Basis & Row Space:

   • A basis must be linearly independent and span the space.
   • Example: For $A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \end{bmatrix}$, the row space basis is {(1,0,2), (0,1,-1)}.

1. Eigenvalues:

   • If a matrix has complex eigenvalues, it’s not diagonalizable over the reals.
   • Symmetric matrices are always diagonalizable.

1. Gradient Descent:

   • Smaller learning rate ($\alpha$) ⇒ slower convergence.
   • Too large $\alpha$ ⇒ divergence.

1. True/False Questions:

   • "Diagonalizable ⇒ Invertible"? False (e.g., zero matrix is diagonalizable but singular).
   • "Real symmetric ⇒ real eigenvalues"? True.
   • "n distinct eigenvalues ⇒ diagonalizable"? True, but the converse is false.