IIT M FOUNDATION AN EXAM QDF3 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 Fundamentals

1. Supervised Learning:

   • Classification (e.g., loan approval: income, credit score → approve/reject).

     • Algorithms: Logistic Regression, Decision Trees, SVM.

   • Regression (predict continuous values).

     • Algorithms: Linear Regression, Polynomial Regression.

   • Loss Functions:

     • Squared Loss (MSE): $L(y, \hat{y}) = (y - \hat{y})^2$.
     • Goal: Minimize loss over training data.

1. Unsupervised Learning:

   • Clustering (e.g., K-Means, Hierarchical Clustering).
   • Dimensionality Reduction (e.g., PCA, SVD).

1. Model Evaluation:

   • Overfitting/underfitting.
   • Train-test split, cross-validation.

B. Linear Algebra (Critical for ML)

1. Matrices & Vectors:

   • Row Space: Span of row vectors of a matrix $A$.

   • Basis: Linearly independent vectors spanning a space.

     • Example: For $A \in \mathbb{R}^{m \times n}$, row space basis has rank $r \leq \min(m, n)$.
   • 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 have real eigenvalues.

   • Diagonalizable Matrix:

     • $A = PDP^{-1}$ (where $D$ is diagonal).
     • Not all diagonalizable matrices are invertible (e.g., singular matrices).

1. Key Theorems:

   • A matrix is diagonalizable iff it has $n$ linearly independent eigenvectors (not necessarily distinct eigenvalues).
   • Spectral Theorem: Real symmetric matrices are diagonalizable.

C. Probability & Statistics (Implied in Questions)

1. Basics:

   • Expected value, variance.
   • Conditional probability (e.g., loan approval given credit score).

1. Bayes’ Theorem (for classification):

D. Calculus (Optimization)

1. Gradient Descent:

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

2. Key Equations

Graph 2: Linear Algebra Concepts

Graph 3: Relationship Between ML and Linear Algebra

Graph 4: Model Evaluation

4. Exam-Specific Insights

1. Classification vs. Regression:

   • Loan approval (Q2) is classification (discrete output: yes/no).

1. Linear Algebra Tricks:

   • Row Space Basis (Q10): Must span all rows and be linearly independent.

     • Correct answer: B (includes all independent rows).
   • Orthogonal Matrix (Q12): $A^T A = I$ (option A).

   • Eigenvalues (Q13, Q15):

     • Real symmetric matrices have real eigenvalues (Q15: True).
     • Non-symmetric matrices may have complex eigenvalues (Q13: A).
   • Diagonalizable ≠ Invertible (Q14: False).

1. Numerical Answers:

   • Q5: 72, Q6: 1024, Q7: 0, Q18: 1 (likely determinant or eigenvalue).

1. True/False:

   • Q14: Diagonalizable ≠ invertible (False).
   • Q15: Real symmetric eigenvalues are real (True).
   • Q16: Diagonalizable ≠ distinct eigenvalues (False; needs full eigenbasis).

5. Study Recommendations

1. Focus Areas:

   • Supervised Learning: Classification (logistic regression, decision trees) vs. regression.
   • Linear Algebra: Row space, basis, eigenvalues, diagonalization.
   • Optimization: Gradient descent, loss functions.

1. Practice:

   • Solve eigenvalue problems (e.g., $2 \times 2$ matrices).
   • Derive squared loss for simple linear regression.
   • Check matrix properties (orthogonal, symmetric) for given examples.

1. Common Pitfalls:

   • Confusing classification with regression.
   • Assuming diagonalizable ⇒ invertible.
   • Forgetting that symmetric matrices have real eigenvalues.