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2024 Oct27: - 27 Oct 2024

MLF Quiz 112 questions40 marks6 MSQ · 4 MCQ · 2 SA
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Q1MSQ3 marksselect all that apply
Which of the following 3 × 3 matrices have rank 2 with column space equal to row space?
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  1. B has eigenvalues -4, 8, 70.
  2. A has eigenvalues 1, 16, 81.
  3. B is an upper triangular matrix.
  4. B is a lower triangular matrix.
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Q5MSQ4 marksselect all that apply
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Q6MSQ4 marksselect all that apply
Determine which of the following limits yields a finite value.
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Q7MCQ3 marks
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  1. Yes
  2. No
  3. Insufficient information
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Q9MCQ3 marks
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  1. 3
  2. 2
  3. -2
  4. 1
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Q11SA3 marks
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Answer key

  1. Q1 (MSQ, 3 marks): A, D
  2. Q2 (MSQ, 3 marks): A, C
  3. Q3 (MSQ, 4 marks): B, C
  4. Q4 (MSQ, 4 marks): A, C, D
  5. Q5 (MSQ, 4 marks): A, F
  6. Q6 (MSQ, 4 marks): A, B, D
  7. Q7 (MCQ, 3 marks): A
  8. Q8 (MCQ, 3 marks): A
  9. Q9 (MCQ, 3 marks): C
  10. Q10 (MCQ, 3 marks): B
  11. Q11 (SA, 3 marks): 1.5
  12. Q12 (SA, 3 marks): 0.4

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

Linear Algebra

  1. Matrices and Rank

    • Rank of a matrix
    • Column space and row space
    • Upper and lower triangular matrices
    • Eigenvalues and eigenvectors
    • Diagonalizable matrices

  1. Matrix Operations

    • Matrix addition and multiplication
    • Matrix inversion
    • Projection matrices

  1. Eigenvalues and Eigenvectors

    • Calculation of eigenvalues and eigenvectors
    • Properties of eigenvalues and eigenvectors

  1. Projection Matrices

    • Projection of a vector onto a line
    • Properties of projection matrices

Calculus

  1. Limits

    • Evaluation of limits
    • Finite and infinite limits

  1. Continuity

    • Continuous functions
    • Conditions for continuity

  1. Derivatives

    • Partial derivatives
    • Directional derivatives
    • Linear approximation

Machine Learning

  1. Least Squares Method

    • Best-fit line
    • Error vectors

  1. Classification Models

    • Misclassification error
    • Sign function

Equations

Linear Algebra

  1. Eigenvalues and Eigenvectors
Av=λvA v = \lambda v

where AA is a matrix, vv is an eigenvector, and λ\lambda is an eigenvalue.

  1. Projection Matrix
P=aaTaTaP = \frac{a a^T}{a^T a}

where aa is a vector.

Calculus

  1. Partial Derivatives
∂f∂x,∂f∂y\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}

for a function f(x,y)f(x, y).

  1. Directional Derivative
Duf(a,b)=∇f(a,b)⋅uD_u f(a, b) = \nabla f(a, b) \cdot u

where ∇f\nabla f is the gradient of ff and uu is a unit vector.

  1. Linear Approximation
L(x,y)=f(a,b)+∂f∂x(a,b)(x−a)+∂f∂y(a,b)(y−b)L(x, y) = f(a, b) + \frac{\partial f}{\partial x}(a, b)(x - a) + \frac{\partial f}{\partial y}(a, b)(y - b)

Machine Learning

  1. Least Squares Method
y^=mx+c\hat{y} = mx + c

where mm is the slope and cc is the intercept.

  1. Sign Function
Sign(x)={1if x≥0−1if x<0\text{Sign}(x) = \begin{cases} 1 & \text{if } x \geq 0 \\ -1 & \text{if } x < 0 \end{cases}