Vector Spaces and Matrix Properties - Linear Algebra - Data Science & Artificial Intelligence Previous Year Questions

1

2024 · Data Science & Artificial Intelligence · Linear Algebra · Vector Spaces and Matrix Properties

Data Science & Artificial Intelligence (DA) 2024

Select all choices that are subspaces of \(\mathbb{R}^3\).
Note: \(\mathbb{R}\) denotes the set of real numbers.

A

\(\left\{ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \mathbb{R}^3 : \mathbf{x} = \alpha \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} + \beta \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \alpha, \beta \in \mathbb{R} \right\}\)

B

\(\left\{ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \mathbb{R}^3 : \mathbf{x} = \alpha^2 \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix} + \beta^2 \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \alpha, \beta \in \mathbb{R} \right\}\)

C

\(\left\{ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \mathbb{R}^3 : 5x_1 + 2x_3 = 0, 4x_1 - 2x_2 + 3x_3 = 0 \right\}\)

D

\(\left\{ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \mathbb{R}^3 : 5x_1 + 2x_3 + 4 = 0 \right\}\)

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2

2024 · Data Science & Artificial Intelligence · Linear Algebra · Vector Spaces and Matrix Properties

Data Science & Artificial Intelligence (DA) 2024

Let \(\mathbb{R}\) be the set of real numbers, \(U\) be a subspace of \(\mathbb{R}^3\) and \(\mathbf{M} \in \mathbb{R}^{3 \times 3}\) be the matrix corresponding to the projection on to the subspace \(U\).
Which of the following statements is/are TRUE?

A

If \(U\) is a 1-dimensional subspace of \(\mathbb{R}^3\), then the null space of \(\mathbf{M}\) is a 1-dimensional subspace.

B

If \(U\) is a 2-dimensional subspace of \(\mathbb{R}^3\), then the null space of \(\mathbf{M}\) is a 1-dimensional subspace.

C

\(\mathbf{M}^2 = \mathbf{M}\)

D

\(\mathbf{M}^3 = \mathbf{M}\)

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3

2025 · Data Science & Artificial Intelligence · Linear Algebra · Vector Spaces and Matrix Properties

Data Science & Artificial Intelligence (DA) 2025

Which of the following statements is/are correct?

A

\(\mathbb{R}^n\) has a unique set of orthonormal basis vectors

B

\(\mathbb{R}^n\) does not have a unique set of orthonormal basis vectors

C

Linearly independent vectors in \(\mathbb{R}^n\) are orthonormal

D

Orthonormal vectors in \(\mathbb{R}^n\) are linearly independent

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4

2025 · Data Science & Artificial Intelligence · Linear Algebra · Vector Spaces and Matrix Properties

Data Science & Artificial Intelligence (DA) 2025

Let \(A \in \mathbb{R}^{n \times n}\) be such that \(A^5 = A\). Which one of the following statements is ALWAYS correct?

A

\(A\) is invertible

B

Determinant of \(A\) is 0

C

The sum of the diagonal elements of \(A\) is 1

D

\(A\) and \(A^5\) have the same rank

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5

2025 · Data Science & Artificial Intelligence · Linear Algebra · Vector Spaces and Matrix Properties

Data Science & Artificial Intelligence (DA) 2025

Let \(\{x_1, x_2, \dots, x_n\}\) be a set of linearly independent vectors in \(\mathbb{R}^n\). Let the \((i, j)\)-th element of matrix \(A \in \mathbb{R}^{n \times n}\) be given by \(A_{ij} = x_i^T x_j\), \(1 \le i, j \le n\). Which one of the following statements is correct?

A

\(A\) is invertible

B

0 is a singular value of \(A\)

C

Determinant of \(A\) is 0

D

\(z^T A z = 0\) for some non-zero \(z \in \mathbb{R}^n\)

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6

2025 · Data Science & Artificial Intelligence · Linear Algebra · Vector Spaces and Matrix Properties

Data Science & Artificial Intelligence (DA) 2025

An \(n \times n\) matrix \(A\) with real entries satisfies the property: \(\|Ax\|^2 = \|x\|^2\), for all \(x \in \mathbb{R}^n\), where \(\|\cdot\|\) denotes the Euclidean norm. Which of the following statements is/are ALWAYS correct?

A

\(A\) must be orthogonal

B

\(A = I\), where \(I\) denotes the identity matrix, is the only solution

C

The eigenvalues of \(A\) are either \(+1\) or \(-1\)

D

\(A\) has full rank

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7

2026 · Data Science & Artificial Intelligence · Linear Algebra · Vector Spaces and Matrix Properties

Data Science and Artificial Intelligence (DA) 2026

Consider a set \( S_1 = \{ x = (x_1, x_2, x_3)^T \in \mathbb{R}^3 \mid x^T x \le 16 \} \). Let \( S_2 \) be another set which is a subspace of \( \mathbb{R}^3 \) with dimension two.
Which of the following gives the area of \( S_1 \cap S_2 \)?

A

\(16\pi \)

B

\(4\pi \)

C

\(4\pi^2 \)

D

\(16\pi^2 \)

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8

2026 · Data Science & Artificial Intelligence · Linear Algebra · Vector Spaces and Matrix Properties

Data Science and Artificial Intelligence (DA) 2026

Let \( M = (I_n - \frac{1}{n} \mathbf{1} \mathbf{1}^T) \) be a matrix, where \( \mathbf{1} = (1,1,1,...,1)^T \in \mathbb{R}^n \) and \( I_n \) is the identity matrix of order \( n \).
Which of the following options is/are correct?

A

\(M^T = M \)

B

\(M^2 = I_n \)

C

\(Trace(M) = n \)

D

\(M \) is a projection matrix

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