IIT M FOUNDATION AN3 EXAM QPF3 16 JULY 2023 - 16 Jul 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. Linear Algebra

• Matrix Operations and Properties

  • Eigenvectors and Eigenvalues
  • Diagonalizable Matrices
  • Row Space, Column Space, and Null Space
  • Orthogonality and Orthonormal Basis

• Key Equations

  • $A \mathbf{x} = \lambda \mathbf{x}$ (Eigenvalue Equation)
  • $A = PDP^{-1}$ (Diagonalization)
  • $\text{Nullspace}(A)$ and $\text{Columnspace}(A)$

2. Regression and Least Squares

• Linear Regression

  • Least Squares Method
  • Projection onto Column Space
  • Error Calculation

• Key Equations

  • $\mathbf{A}^T \mathbf{A} \mathbf{x} = \mathbf{A}^T \mathbf{b}$ (Normal Equation)
  • $\text{Error} = \|\mathbf{b} - \mathbf{A}\mathbf{x}\|^2$

3. Calculus and Optimization

• Gradient and Directional Derivatives

  • Gradient Descent Algorithm
  • Directional Derivative Calculation

• Key Equations

  • $\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$
  • $X_{n+1} = X_n - h \nabla f(X_n)$ (Gradient Descent Update)

4. Dimensionality Reduction

• Encoder and Decoder Functions

  • Reconstruction Error Calculation

• Key Equations

  • $R(f, g) = \frac{1}{n} \sum_{i=1}^{n} \| x_i - g(f(x_i)) \|^2$

5. Continuity and Piecewise Functions

• Continuity Conditions

  • Piecewise Function Definition

• Key Equations

  • $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)$ (Continuity Condition)

6. Projection Matrices

• Projection onto a Line and Plane

  • Projection Matrix Properties

• Key Equations

  • $P_1$ and $P_2$ (Projection Matrices)
  • $\det(P_1 + P_2)$