cs2004_2026T2_Q1_NA.pdf

Machine Learning Foundations(MLF) · Quiz 1 · May 2026

Questions and published explanations below are available without starting a test. Some questions may not have a published solution yet.

Question 2 MCQ · 3.0 marks

What is the best linear approximation of [[IMAGE:ce7276dd7a29fbd0_2_2]] around [[IMAGE:ce7276dd7a29fbd0_2_3]]

[[IMAGE:ce7276dd7a29fbd0_2_4]]
[[IMAGE:ce7276dd7a29fbd0_2_5]]
[[IMAGE:ce7276dd7a29fbd0_2_6]]
[[IMAGE:ce7276dd7a29fbd0_2_7]]

A published solution is not available for this question yet.

Question 3 MCQ · 3.0 marks

A company manufactures a special rectangular metal sheet. The production cost (in hundreds of rupees) depends on its length [[IMAGE:ce7276dd7a29fbd0_2_8]] meters and width [[IMAGE:ce7276dd7a29fbd0_2_9]] meters according to [[IMAGE:ce7276dd7a29fbd0_2_10]] . At a particular stage, the dimensions of the sheet are [[IMAGE:ce7276dd7a29fbd0_2_11]] . The company plans to change the dimensions in the direction represented by the vector [[IMAGE:ce7276dd7a29fbd0_2_12]] and wants to determine how rapidly the production cost changes in that specific direction. Find the directional derivative of [[IMAGE:ce7276dd7a29fbd0_3_13]] at the point [[IMAGE:ce7276dd7a29fbd0_3_14]] in the direction of [[IMAGE:ce7276dd7a29fbd0_3_15]] .

[[IMAGE:ce7276dd7a29fbd0_3_16]]
[[IMAGE:ce7276dd7a29fbd0_3_17]]
[[IMAGE:ce7276dd7a29fbd0_3_18]]
[[IMAGE:ce7276dd7a29fbd0_3_19]]

A published solution is not available for this question yet.

Question 4 MCQ · 3.0 marks

Find the projection matrix that projects the [[IMAGE:ce7276dd7a29fbd0_3_20]] plane onto the line [[IMAGE:ce7276dd7a29fbd0_3_21]] .

[[IMAGE:ce7276dd7a29fbd0_3_22]]
[[IMAGE:ce7276dd7a29fbd0_3_23]]
[[IMAGE:ce7276dd7a29fbd0_3_24]]
[[IMAGE:ce7276dd7a29fbd0_3_25]]

A published solution is not available for this question yet.

Question 5 MCQ · 3.0 marks

Let [[IMAGE:ce7276dd7a29fbd0_3_26]] be a real symmetric matrix. Consider the following statements. (1) [[IMAGE:ce7276dd7a29fbd0_3_27]] is orthogonally diagonalizable. (2) [[IMAGE:ce7276dd7a29fbd0_4_28]] is not diagonalizable. (3) All the eigenvalues of [[IMAGE:ce7276dd7a29fbd0_4_29]] are real. (4) The eigenvalues of [[IMAGE:ce7276dd7a29fbd0_4_30]] may be imaginary. (5) Eigenvectors corresponding to distinct eigenvalues of [[IMAGE:ce7276dd7a29fbd0_4_31]] are linearly independent. (6) Eigenvectors corresponding to distinct eigenvalues of [[IMAGE:ce7276dd7a29fbd0_4_32]] are linearly dependent. Which of the above statements are true?

(1), (3), (5)
(2), (4), (6)
(2), (3), (5)
(1), (4), (6)

A published solution is not available for this question yet.

Question 6 MSQ · 3.0 marks

Let [[IMAGE:ce7276dd7a29fbd0_4_33]] and [[IMAGE:ce7276dd7a29fbd0_4_34]] be two square matrices of order [[IMAGE:ce7276dd7a29fbd0_4_35]] . The rows of [[IMAGE:ce7276dd7a29fbd0_4_36]] are represented by row vectors [[IMAGE:ce7276dd7a29fbd0_4_37]] , from top to bottom, and the columns of [[IMAGE:ce7276dd7a29fbd0_4_38]] are represented by column vectors [[IMAGE:ce7276dd7a29fbd0_4_39]] , from left to right. The second row of [[IMAGE:ce7276dd7a29fbd0_4_40]] is [[IMAGE:ce7276dd7a29fbd0_4_41]] . Which of the following is/are true?

The second row of [[IMAGE:ce7276dd7a29fbd0_4_42]] is [[IMAGE:ce7276dd7a29fbd0_4_43]] .
The second column of [[IMAGE:ce7276dd7a29fbd0_4_44]] is [[IMAGE:ce7276dd7a29fbd0_4_45]] .
The row space of [[IMAGE:ce7276dd7a29fbd0_4_46]] is contained in the row space of [[IMAGE:ce7276dd7a29fbd0_4_47]] .
The row space of [[IMAGE:ce7276dd7a29fbd0_4_48]] is contained in the row space of [[IMAGE:ce7276dd7a29fbd0_4_49]] .

A published solution is not available for this question yet.

Question 7 MSQ · 3.0 marks

[[IMAGE:ce7276dd7a29fbd0_5_50]] is a square matrix of order [[IMAGE:ce7276dd7a29fbd0_5_51]] with the eigenvalue [[IMAGE:ce7276dd7a29fbd0_5_52]] repeated twice. Which of the following is/are true?

[[IMAGE:ce7276dd7a29fbd0_5_53]] is invertible.
Each of the entries on the main diagonal of [[IMAGE:ce7276dd7a29fbd0_5_54]] is equal to [[IMAGE:ce7276dd7a29fbd0_5_55]] .
[[IMAGE:ce7276dd7a29fbd0_5_56]] has two linearly independent eigenvectors.
[[IMAGE:ce7276dd7a29fbd0_5_57]] is not invertible, where [[IMAGE:ce7276dd7a29fbd0_5_58]] is the identity matrix.

A published solution is not available for this question yet.

Question 8 MSQ · 4.0 marks

Suppose [[IMAGE:ce7276dd7a29fbd0_5_59]] is a [[IMAGE:ce7276dd7a29fbd0_5_60]] matrix with rank 4. Which of the following is/are true?

Columns of [[IMAGE:ce7276dd7a29fbd0_5_61]] are linearly independent.
Null space of [[IMAGE:ce7276dd7a29fbd0_5_62]] contains only the zero vector.
Row space of [[IMAGE:ce7276dd7a29fbd0_5_63]] is [[IMAGE:ce7276dd7a29fbd0_5_64]] .
Left null space of [[IMAGE:ce7276dd7a29fbd0_5_65]] has dimension 2.

A published solution is not available for this question yet.

Question 9 MSQ · 4.0 marks

Which of the following represents the set of all points on the line passing through [[IMAGE:ce7276dd7a29fbd0_5_66]] along the direction [[IMAGE:ce7276dd7a29fbd0_5_67]] ? (More than one option can be correct).

[[IMAGE:ce7276dd7a29fbd0_6_68]]
[[IMAGE:ce7276dd7a29fbd0_6_69]]
[[IMAGE:ce7276dd7a29fbd0_6_70]]
[[IMAGE:ce7276dd7a29fbd0_6_71]]

A published solution is not available for this question yet.

Question 10 NAT · 3.0 marks

Let [[IMAGE:ce7276dd7a29fbd0_6_72]] be two vectors with [[IMAGE:ce7276dd7a29fbd0_6_73]] and [[IMAGE:ce7276dd7a29fbd0_6_74]] . Find the maximum possible value of [[IMAGE:ce7276dd7a29fbd0_6_75]] , where [[IMAGE:ce7276dd7a29fbd0_6_76]] represents the absolute value of [[IMAGE:ce7276dd7a29fbd0_6_77]] . Your answer should be an integer.

A published solution is not available for this question yet.

Question 11 NAT · 4.0 marks

[[IMAGE:ce7276dd7a29fbd0_6_78]] Let be the matrix whose eigenvalues are [[IMAGE:ce7276dd7a29fbd0_6_79]] and [[IMAGE:ce7276dd7a29fbd0_6_80]] , and whose eigenvectors are [[IMAGE:ce7276dd7a29fbd0_6_81]] and [[IMAGE:ce7276dd7a29fbd0_6_82]] respectively. Find [[IMAGE:ce7276dd7a29fbd0_6_83]] . Your answer should be an integer.

A published solution is not available for this question yet.

Question 12 NAT · 3.0 marks

Consider the following dataset consisting of two features [[IMAGE:ce7276dd7a29fbd0_7_84]] and [[IMAGE:ce7276dd7a29fbd0_7_85]] : [[IMAGE:ce7276dd7a29fbd0_7_86]] We want to reduce the dimensionality of the dataset from [[IMAGE:ce7276dd7a29fbd0_7_87]] to [[IMAGE:ce7276dd7a29fbd0_7_88]] using the following encoder- decoder pairs. [[IMAGE:ce7276dd7a29fbd0_7_89]] [[IMAGE:ce7276dd7a29fbd0_7_90]] The reconstruction loss is defined as the mean squared distance between the reconstructed input and the original input. Based on the above information, answer the given sub-questions.

Compute the reconstruction loss for Pair 1. **NOTE:** Your answer should be an integer