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IIT M FOUNDATION AN4 EXAM QPF4 16 JULY 2023 - 16 Jul 2023

MLF Quiz 115 questions50 marks6 MSQ · 6 SA · 3 MCQFoundation / QPF4
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Q1MSQ3 marksselect all that apply
Which of the following statements are correct?
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Q2MSQ3 marksselect all that apply
Which of the following statements is/are true?
  1. Row space of a matrix is orthogonal to the null space of the matrix.
  2. Row space of a matrix is orthogonal to the null space of transpose of thematrix.
  3. Dimension of row space of a matrix is always equal to the dimension ofcolumn space of the matrix.
  4. Dimension of row space of a matrix need not be equal to the dimension ofcolumn space of the matrix.
Q3MSQ3 marksselect all that apply
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Q4MSQ3 marksselect all that apply
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Q5MSQ4 marksselect all that apply
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Q6MSQ4 marksselect all that apply
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Q7MCQ4 marks
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Q8MCQ4 marks
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  1. (1, 3)
  2. (1/2, 1/6)
  3. (0, 0)
  4. The algorithm does not converge
Q9SA4 marks
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Q10SA4 marks
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Q11SA4 marks
Let P1 be the projection matrix onto the line spanned by the vector a = [1, 2, 3]T . Let P2 be the projection matrix that projects vectors onto the plane perpendicular to span of a. Find the determinant of P1 + P2.
Q12SA3 marks
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Q13SA2 marks
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Comprehension passage - the questions below it refer to this
Based on the above data, answer the given subquestions. question figure
Q14MCQ2 marks
Is the system Ax = b consistent?
  1. Yes
  2. No
Q15SA3 marks
Using the least square approximation, find the length of the projection vector p of b onto the column space of A. Enter the answer correct to two decimal places.

Answer key

  1. Q1 (MSQ, 3 marks): A, D
  2. Q2 (MSQ, 3 marks): A, C
  3. Q3 (MSQ, 3 marks): A, B
  4. Q4 (MSQ, 3 marks): B, D
  5. Q5 (MSQ, 4 marks): A, B, C
  6. Q6 (MSQ, 4 marks): A, C
  7. Q7 (MCQ, 4 marks): A
  8. Q8 (MCQ, 4 marks): C
  9. Q9 (SA, 4 marks): 5
  10. Q10 (SA, 4 marks): 37 to 38
  11. Q11 (SA, 4 marks): 1
  12. Q12 (SA, 3 marks): -1.16 to -1.12
  13. Q13 (SA, 2 marks): 3.5
  14. Q14 (MCQ, 2 marks): B
  15. Q15 (SA, 3 marks): 3.12 to 3.16

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

Mathematics

  1. Linear Algebra

    • Matrix Operations: Understanding matrix multiplication, inversion, and properties.

    • Eigenvalues and Eigenvectors: Key concepts for understanding matrix diagonalization and properties.

      • Av=λvA v = \lambda v

    • Orthogonality: Row space and null space orthogonality.

      • Row space(A)⊥Null space(A)\text{Row space}(A) \perp \text{Null space}(A)

    • Projection Matrices: Projection onto lines and planes.

      • P1P_1 and P2P_2 for projection onto a line and its perpendicular plane.

    • Least Squares Method: Solving linear regression problems.

      • ATAx=ATbA^T A x = A^T b

    • Consistency of Linear Systems: Determining if a system Ax=bAx = b has a solution.

      • Rank(A)=Rank([A∣b])\text{Rank}(A) = \text{Rank}([A|b])

  1. Calculus

    • Directional Derivatives: Calculating derivatives in specific directions.

      • Dn^f=∇f⋅n^D_{\hat{n}} f = \nabla f \cdot \hat{n}

    • Gradient Descent: Optimization algorithm for minimizing functions.

      • Xn+1=Xn−h∇f(Xn)X_{n+1} = X_n - h \nabla f(X_n)

    • Continuity of Functions: Ensuring functions are continuous across their domain.

      • lim⁡x→cf(x)=f(c)\lim_{x \to c} f(x) = f(c)

  1. Statistics and Probability

    • Gaussian Mixture Models: Modeling multimodal data.

    • Loss Functions: Mean Squared Error (MSE) and other loss functions.

      • L=max⁡i∣f(xi)−yi∣L = \max_i |f(x^i) - y^i|

    • Reconstruction Error: Error in dimensionality reduction.

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

Data Science

  1. Linear Regression: Fitting models to data.

    • f(X)=wTxf(X) = w^T x

  2. Dimensionality Reduction: Techniques like PCA and autoencoders.

    • Encoder: f(x1,x2,x3)=x1+x2+x32f(x_1, x_2, x_3) = \frac{x_1 + x_2 + x_3}{2}
    • Decoder: g(u)=[u,u,u]g(u) = [u, u, u]

Software Engineering

  1. Algorithm Implementation: Understanding and implementing algorithms like gradient descent.