IIT M DIPLOMA AN2 EXAM QPD2 29 Oct 2023 - 29 Oct 2023

Concepts this paper tests

Auto-generated summary from the source site. Handy as a checklist; the lessons above explain each idea from scratch.

Summary of Core Topics, Concepts, Principles, and Equations

1. Supervised Learning

• Definition: Learning from labeled data.
• Tasks: Classification and regression.
• Examples: Predicting weather conditions, identifying objects in images, categorizing news articles.

2. Misclassification Error

• Definition: Error rate of a classification model.
• Models: Sign functions of linear combinations of features.
• Example: Sign(x1 - x2) for given data points.

3. Column Space

• Definition: The set of all linear combinations of the columns of a matrix.
• Example: Checking if a vector belongs to the column space of a matrix.

4. Diagonalization of Matrices

• Definition: Process of finding a matrix $P$ such that $P^{-1}AP$ is diagonal.
• Steps: Find eigenvalues and eigenvectors.
• Example: Diagonalizing a 2x2 matrix.

5. Mean Squared Loss

• Definition: Average squared difference between predicted and actual values.
• Formula: $\frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2$
• Example: Calculating mean squared loss for a regression model.

6. Quadratic Curves and Tangents

• Definition: Curve fitting and tangency conditions.
• Example: Finding the product $ab$ for a quadratic curve tangent to a line.

7. Directional Derivatives

• Definition: Rate of change of a function in a specific direction.
• Formula: $\nabla f \cdot \mathbf{u}$
• Example: Calculating directional derivatives at a point.

8. Eigenvalues and Eigenvectors

• Definition: Vectors that change by a scalar factor when a linear transformation is applied.
• Example: Finding the sum of squares of entries of a matrix given its eigenvectors and eigenvalues.

9. Differentiability and Continuity

• Definition: Conditions for a function to be differentiable and continuous.
• Example: Analyzing a piecewise function for differentiability and continuity.

10. Second Order Approximation

• Definition: Approximating a function using its Taylor series expansion up to the second order.
• Formula: $f(x, y) \approx f(a, b) + f_x(a, b)(x-a) + f_y(a, b)(y-b) + \frac{1}{2} [f_{xx}(a, b)(x-a)^2 + 2f_{xy}(a, b)(x-a)(y-b) + f_{yy}(a, b)(y-b)^2]$
• Example: Second order approximation for $e^{2y - x}$.

11. Least Squares Method

• Definition: Method for finding the best fit line for a dataset.
• Formula: Minimizing the sum of squared residuals.
• Example: Finding the best fit line for a given dataset.

12. Matrix Properties

• Definitions: Invertibility, symmetry, rank, diagonalizability.
• Examples: Statements about matrix properties and their validity.

13. Vector Spaces and Projections

• Definitions: Row space, column space, null space, projection onto a subspace.
• Examples: Finding the projection of a vector onto a subspace and the length of the error vector.