IIT M IMPROVEMENT AN EXAM QIA4 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.

Core Topics & Concepts

1. Machine Learning Foundations

A. Supervised Learning

• Classification vs. Regression:

  • Classification: Predict discrete labels (e.g., loan approval: Yes/No).

    • Algorithms: Logistic Regression, Decision Trees, SVM.

  • Regression: Predict continuous values (e.g., house price).

    • Algorithms: Linear Regression, Polynomial Regression.
  • Key Equation (Polynomial Regression):

• Least Squares Method: Minimize squared error:

B. Model Evaluation

• Loss Functions:

  • Squared Loss (MSE):

• Reconstruction Loss (Autoencoders):

where $f$ = encoder, $g$ = decoder.

C. Linear Algebra for ML

• Vectors & Orthogonality:

  • Two vectors $v_1, v_2$ are orthogonal if $v_1^T v_2 = 0$.
  • Projection of $v_1$ onto $v_2$:

• Linear Independence/Dependence:

  • Vectors $\{v_1, v_2, \dots, v_n\}$ are linearly independent if no vector can be written as a linear combination of others.
  • Span: The set of all linear combinations of vectors.

• Eigenvalues/Eigenvectors:

  • For matrix $A$, if $Av = \lambda v$, then $\lambda$ is an eigenvalue and $v$ is an eigenvector.
  • Determinant: Product of eigenvalues.

• Diagonalizable Matrix: $A = PDP^{-1}$ where $D$ is a diagonal matrix of eigenvalues.
• Symmetric Matrices: All eigenvalues are real.
• Orthogonal Matrices: $A^T A = I$ (eigenvalues have magnitude 1).

D. Dimensionality Reduction

• Autoencoders:

  • Encoder $f: \mathbb{R}^n \rightarrow \mathbb{R}^k$ (compression).
  • Decoder $g: \mathbb{R}^k \rightarrow \mathbb{R}^n$ (reconstruction).
  • Loss: Mean squared error between input and reconstruction.

E. Basis & Row Space

• Basis: Linearly independent vectors that span a space.
• Row Space: Span of the rows of a matrix.
• Rank: Dimension of the row/column space.

2. Linear Algebra Deep Dive

A. Matrix Properties

• Invertibility: A matrix is invertible iff $\det(A) \neq 0$.

• Eigenvalue Theorems:

  • If the sum of columns of $A$ is the zero vector, then $\lambda = 0$ is an eigenvalue.
  • A matrix with $n$ distinct eigenvalues is diagonalizable.

• Special Matrices:

  • Orthogonal Matrix: $A^T A = I$ (preserves norms, eigenvalues lie on unit circle).
  • Rotation Matrix:

• Eigenvalues: $e^{i\theta}, e^{-i\theta}$ (complex if $\theta \neq 0, \pi$).

B. Linear Approximation

• First-Order Taylor Approximation (for $f: \mathbb{R}^n \rightarrow \mathbb{R}$):

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

• Tangent Plane Equation:

3. Key Equations Summary

Graph 2: Linear Algebra Concepts

Graph 3: Calculus for ML

Graph 4: Eigenvalues & Special Matrices

Exam Tips

1. Polynomial Regression: Always use least squares to solve for $\theta$ (normal equations or gradient descent).
2. Orthogonality/Projection: Check dot product for orthogonality; projection formula is critical.

3. Eigenvalues:

   • Sum of columns = zero vector $\Rightarrow \lambda = 0$.
   • Rotation matrices have complex eigenvalues (unless $\theta = 0, \pi$).
4. Linear Approximation: Compute gradient and evaluate at the point.
5. Basis/Row Space: Row reduce the matrix to find basis vectors.
6. Autoencoders: Reconstruction loss is MSE between input and output.