IIT M IMPROVEMENT AN EXAM QIB3 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 (e.g., loan approval: B in Q2).

     • Binary/multi-class classification.
     • Algorithms: Logistic Regression, Decision Trees, SVM.

   • Regression (e.g., polynomial regression: Q3, Q8, Q9).

     • Linear/non-linear regression (e.g., quadratic model: $y = \theta_0 + \theta_1 x + \theta_2 x^2$).

     • Loss Functions:

       • Squared loss (MSE): $L = \frac{1}{n}\sum_{i=1}^n (y_i - \hat{y}_i)^2$ (Q5, Q8).
       • Reconstruction loss (autoencoders): Mean squared distance between input and reconstructed input (Q5).

   • Model Evaluation:

     • Overfitting/underfitting.
     • Zero loss implies perfect fit (only possible if data is linearly separable; Q9).

1. Unsupervised Learning:

   • Clustering (e.g., K-means, hierarchical).
   • Dimensionality Reduction (e.g., PCA, autoencoders; Q5).

1. Key Algorithms:

   • Least Squares Method (for regression; Q3):

     • Solve $\theta = (X^T X)^{-1} X^T y$ for linear regression.
     • For polynomial regression, transform $X$ into a matrix of powers (e.g., $[1, x, x^2]$ for quadratic).

   • Autoencoders (Q5):

     • Encoder $f$, decoder $g$, reconstruction loss.

B. Linear Algebra

1. Vectors:

   • Orthogonality: $v_1 \cdot v_2 = 0$ (Q4).

   • Projection: Projection of $v_1$ onto $v_2$: $\text{proj}_{v_2} v_1 = \left( \frac{v_1 \cdot v_2}{v_2 \cdot v_2} \right) v_2$.

     • Zero projection implies orthogonality (Q4: options B, C, D).

1. Matrices:

   • Eigenvalues/Eigenvectors (Q6, Q7, Q13–Q16):

     • $A v = \lambda v$.

     • Properties:

       • Sum of columns = zero vector ⇒ $\lambda = 0$ is an eigenvalue (Q7).
       • Real symmetric matrices have real eigenvalues (Q15).
       • Diagonalizable ≠ invertible (Q14).
       • Diagonalizable iff $n$ linearly independent eigenvectors (not necessarily distinct eigenvalues; Q16).

   • Special Matrices:

     • Orthogonal Matrix: $A^T A = I$ (Q12, Q13).

       • Rotation matrix: $A = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$ has no real eigenvalues for $0 < \theta < 180^\circ$ (Q13).
     • Determinant: $\det(A)$ for scalar matrices (Q6: $A = 2I$ ⇒ $\det(A) = 2^{10} = 1024$).

   • Row Space/Basis (Q10, Q11):

     • Row space: Span of rows of $A$.
     • Basis: Linearly independent rows that span the row space.
     • Linear relations among rows (Q11: $R_2 = 2R_1 + R_3$).

1. Systems of Equations:

   • Collinearity check for linear models (Q9: slopes must be equal for all points).

C. Calculus & Optimization

1. Linear Approximation (Q17, Q18):

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

     • $L_{(0,0)}[f](x) = f(0,0) + \nabla f(0,0) \cdot x = 1 + x_1 + x_2$ (Q17).
   • Plane Equation: $w^T x + b = 0$, where $w = \nabla f$, $b = -f(0,0)$ (Q18: $b = 1$).

1. Contour Plots (Q19):

   • Contours of $f(x_1, x_2) = e^{x_1} + x_2$ are horizontal curves (since $f$ increases monotonically with $x_1$ and $x_2$).

D. Probability & Statistics (Not directly tested here, but often paired with ML)

• Not covered in this paper.

2. Key Equations

Graph 2: Linear Algebra Concepts

Graph 3: Calculus & Optimization

Graph 4: Relationships Between Topics