IIT M IMPROVEMENT AN EXAM QIB2 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 essential for the exam, followed by Mermaid knowledge graphs to visualize relationships.

1. Core Topics & Concepts

A. Machine Learning Foundations

1. Supervised Learning

• Classification vs. Regression:

  • Classification: Predict discrete labels (e.g., loan approval: Yes/No). Example: Q2 (Loan approval based on income/credit score → Classification).
  • Regression: Predict continuous values (e.g., house price, temperature). Example: Q3 (Polynomial regression for $y = \theta_0 + \theta_1 x + \theta_2 x^2$).

• Loss Functions:

  • Squared Loss (MSE): $L = \frac{1}{n} \sum_{i=1}^n (y_i - \hat{y}_i)^2$. Example: Q5 (Reconstruction loss in autoencoders), Q8 (Choosing model with smallest squared loss).

• Polynomial Regression:

  • Fit a quadratic model using least squares:

Example: Q3 (Fit $y = \theta_0 + \theta_1 x + \theta_2 x^2$ to given data).

• Linear Approximation:

  • First-order Taylor expansion for $f(x)$ around $(a, b)$:

Example: Q17–19 (Linear approximation of $f(x_1, x_2) = e^{x_1} + x_2$ at $(0,0)$).

2. Unsupervised Learning

• Autoencoders:

  • Encoder: $f(x)$ compresses input to latent space $u$.
  • Decoder: $g(u)$ reconstructs input from $u$.
  • Reconstruction Loss: Mean squared error between input and output. Example: Q5 (Compute loss for $f(x_1, x_2, x_3) = 0.5x_1 + 0.5x_2 - x_3$ and $g(u) = [u, u, u]$).

B. Linear Algebra

1. Vectors & Orthogonality

• Orthogonal Vectors: Dot product $v_1 \cdot v_2 = 0$. Example: Q4 (Check if $v_1 = (1,2,2)^T$ and $v_2 = (2,-1,0)^T$ are orthogonal).
• Linear Independence: Vectors $\{v_1, v_2, \dots, v_n\}$ are independent if $c_1 v_1 + \dots + c_n v_n = 0$ implies all $c_i = 0$.
• Span: Set of all linear combinations of vectors. Two vectors cannot span $\mathbb{R}^3$. Example: Q4 (Options B and C).

2. Matrices

• Eigenvalues/Vectors:

  • $A v = \lambda v$. If all non-zero vectors are eigenvectors, $A$ is a scalar matrix ($A = \lambda I$). Example: Q6 ($A = 2I_{10}$, so $\det(A) = 2^{10} = 1024$).
  • Sum of columns is zero vector → $\lambda = 0$ is an eigenvalue. Example: Q7.

• Diagonalizable Matrices:

  • $A$ is diagonalizable iff it has $n$ linearly independent eigenvectors.
  • Not necessarily requires $n$ distinct eigenvalues (e.g., identity matrix). Example: Q16 (False: "A is diagonalizable iff it has $n$ distinct eigenvalues").

• Orthogonal Matrices:

  • $A^T A = I$. Preserves norms/angles. Example: Q12–13 (Rotation matrix $A = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$ is orthogonal and has no real eigenvalues for $0 < \theta < 180^\circ$).

• Row Space & Basis:

  • Row space = span of rows. Basis is a linearly independent set spanning the row space. Example: Q10 (Basis for row space of $A = \begin{bmatrix} 1 & 0 & 2 \\ 3 & 1 & 5 \\ 2 & -1 & 1 \end{bmatrix}$ is $\{(1,0,2), (0,1,-1)\}$).

3. Systems of Equations

• Linear Relations:

  • Check if rows/columns are linearly dependent. Example: Q11 ($R_2 = 2R_1 + R_3$ is false; correct relation is $0.9R_1 + 0.8R_2 + 0.5R_3 = 0$).

C. Calculus & Optimization

1. Linear Approximation

• For $f: \mathbb{R}^2 \to \mathbb{R}$, the linear approximation at $(a,b)$ is:

Example: Q17–18 (For $f(x_1, x_2) = e^{x_1} + x_2$ at $(0,0)$, $L_{(0,0)}[f](x) = 1 + x_1 + x_2$).

2. Plane Equations

• Plane normal to $w \in \mathbb{R}^3$: $w^T x + b = 0$. Example: Q18 (Given $w^T w = 3$, find $b = 1$).

2. Key Equations

Graph 2: Linear Algebra Concepts

Graph 3: Calculus & Optimization

4. Exam Tips

1. Classification vs. Regression: Identify the problem type (discrete/continuous output).
2. Polynomial Regression: Use least squares to solve for $\theta$.

3. Eigenvalues:

   • If all non-zero vectors are eigenvectors → $A = \lambda I$.
   • Sum of columns is zero → $\lambda = 0$ is an eigenvalue.
4. Orthogonal Matrices: Rotation matrices are orthogonal but may lack real eigenvalues.
5. Linear Approximation: Compute gradient and evaluate at the point.
6. Basis: Ensure vectors are linearly independent and span the space.