IIT M IMPROVEMENT AN EXAM QIM3 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 (Computer-Based 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).
   • Regression (e.g., predicting continuous values).

   • Key Algorithms:

     • Linear Regression (Least Squares).
     • Polynomial Regression (e.g., quadratic model: $y = \theta_0 + \theta_1 x + \theta_2 x^2$).

   • Loss Functions:

     • Squared Loss: $L = \sum (y_i - \hat{y}_i)^2$.
     • Reconstruction Loss (Autoencoders): Mean squared distance between input and reconstructed input.

1. Unsupervised Learning:

   • Clustering (grouping similar data points).
   • Dimensionality Reduction (e.g., PCA, Autoencoders).

1. Model Evaluation:

   • Overfitting/Underfitting: Check if a linear model can achieve zero loss (e.g., Q9: No, because points are not colinear).

B. Linear Algebra

1. Vectors:

   • Orthogonality: Two vectors $v_1$ and $v_2$ are orthogonal if $v_1 \cdot v_2 = 0$.

   • Linear Dependence/Independence:

     • Vectors are linearly dependent if one is a scalar multiple of the other.
     • Basis: A set of linearly independent vectors spanning a space (e.g., row space of a matrix).
   • Projection: Projection of $v_1$ onto $v_2$: $\text{proj}_{v_2} v_1 = \left( \frac{v_1 \cdot v_2}{v_2 \cdot v_2} \right) v_2$.

1. Matrices:

   • Eigenvalues/Eigenvectors:

     • $A v = \lambda v$.
     • If the sum of columns of $A$ is the zero vector, $\lambda = 0$ is an eigenvalue.
     • Diagonalizable Matrices: A matrix is diagonalizable if it has $n$ linearly independent eigenvectors (not necessarily distinct eigenvalues).
     • Real Symmetric Matrices: Always have real eigenvalues.

   • Orthogonal Matrices:

     • $A^T A = I$.
     • Example: Rotation matrix $A = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$.

     • Properties:

       • Preserves vector lengths ($\|A v\| = \|v\|$).
       • Eigenvalues: $e^{i\theta}$ and $e^{-i\theta}$ (complex if $\theta \neq 0, \pi$).

   • Determinant:

     • For a matrix where every non-zero vector is an eigenvector with eigenvalue $\lambda$, $A = \lambda I$, so $\det(A) = \lambda^n$.
     • Example: $10 \times 10$ matrix with eigenvalue $2$ for all vectors → $\det(A) = 2^{10} = 1024$.

1. Systems of Equations:

   • Row Space: Spanned by the rows of $A$.
   • Linear Relations: e.g., $R_2 = 2R_1 + R_3$.

C. Calculus & Optimization

1. Linear Approximation:

   • For $f(x_1, x_2) = e^{x_1} + x_2$, the linear approximation at $(0,0)$ is:

• Gradient: $\nabla f = (e^{x_1}, 1)$.
• Plane Equation: $w^T x + b = 0$, where $w = \nabla f$ and $b = -f(0,0)$.

1. Contour Plots:

   • For $f(x_1, x_2) = e^{x_1} + x_2$, contours are horizontal shifts of the exponential function.

D. Key Equations

Graph 2: Linear Algebra Concepts

Graph 3: Calculus & Optimization

Graph 4: Relationships Between Topics

3. Exam Preparation Tips

1. Focus Areas:

   • Polynomial Regression: Know how to fit a quadratic model using least squares.
   • Eigenvalues/Vectors: Understand conditions for diagonalizability, real symmetric matrices, and determinants.
   • Linear Approximation: Compute gradients and plane equations.
   • Orthogonality: Check dot products and projections.
   • Basis/Row Space: Find bases for row spaces and linear relations between rows.

1. Practice Problems:

   • Solve for $\theta$ in $y = \theta_0 + \theta_1 x + \theta_2 x^2$ using least squares.
   • Compute eigenvalues for simple matrices (e.g., rotation matrices).
   • Derive linear approximations for multivariate functions.
   • Verify orthogonality and linear independence of vectors.

1. Common Pitfalls:

   • Confusing classification (discrete output) with regression (continuous output).
   • Misapplying eigenvalue conditions (e.g., assuming all matrices with repeated eigenvalues are diagonalizable).
   • Incorrectly computing projections or dot products.