GATE DA 2024 | Question 47

Select all choices that are subspaces of \(\mathbb{R}^{3}\).

Note: \(\mathbb{R}\) denotes the set of real numbers.

Option A: \(\left\{\mathbf{x} = \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix} \in \mathbb{R}^{3} : \mathbf{x} = \alpha \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} + \beta \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \alpha, \beta \in \mathbb{R}\right\}\) (correct)
Option B: \(\left\{\mathbf{x} = \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix} \in \mathbb{R}^{3} : \mathbf{x} = \alpha^{2} \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix} + \beta^{2} \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \alpha, \beta \in \mathbb{R}\right\}\)
Option C: \(\left\{\mathbf{x} = \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix} \in \mathbb{R}^{3} : 5x_{1} + 2x_{3} = 0, 4x_{1} - 2x_{2} + 3x_{3} = 0\right\}\) (correct)
Option D: \(\left\{\mathbf{x} = \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix} \in \mathbb{R}^{3} : 5x_{1} + 2x_{3} + 4 = 0\right\}\)

Answer

(A), (C)

A subset of \(\mathbb{R}^{3}\) is a subspace when it contains \(\mathbf{0}\) and is closed under addition and scalar multiplication.

(A) is the span of \((1, 1, 0)\) and \((1, 0, 0)\), the plane \(x_{3} = 0\): a subspace.
(B) The coefficients \(\alpha^{2}\) and \(\beta^{2}\) are never negative, so the set holds only the non-negative combinations of the two vectors (a cone, not a plane). It contains \((1, 2, 0)\) but not \((-1, -2, 0)\), which would need \(\alpha^{2} = -1\).
(C) is the solution set of a homogeneous linear system, the null space of a \(2 \times 3\) matrix: a subspace.
(D) The zero vector gives \(5(0) + 2(0) + 4 = 4 \ne 0\), so \(\mathbf{0}\) is missing; this is a shifted plane.

Answer: A, C