The GATE Grind

GATE 2024 DA – Question 47

Linear Algebra · Vector spaces, linear independence, rank and nullity · 2 marks · Multiple select

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

  1. $\{\mathbf{x} \in \mathbb{R}^3 : \mathbf{x} = \alpha(1, 1, 0)^T + \beta(1, 0, 0)^T, \alpha, \beta \in \mathbb{R}\}$
  2. $\{\mathbf{x} \in \mathbb{R}^3 : \mathbf{x} = \alpha^2(1, 2, 0)^T + \beta^2(1, 0, 1)^T, \alpha, \beta \in \mathbb{R}\}$
  3. $\{\mathbf{x} \in \mathbb{R}^3 : 5x_1 + 2x_3 = 0, 4x_1 - 2x_2 + 3x_3 = 0\}$
  4. $\{\mathbf{x} \in \mathbb{R}^3 : 5x_1 + 2x_3 + 4 = 0\}$

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Show answer and explanation

Correct answer: (A) $\{\mathbf{x} \in \mathbb{R}^3 : \mathbf{x} = \alpha(1, 1, 0)^T + \beta(1, 0, 0)^T, \alpha, \beta \in \mathbb{R}\}$; (C) $\{\mathbf{x} \in \mathbb{R}^3 : 5x_1 + 2x_3 = 0, 4x_1 - 2x_2 + 3x_3 = 0\}$

Explanation

A is the span of two vectors, so it is a subspace. In B the coefficients $\alpha^2$ and $\beta^2$ are never negative, so the set is not closed under multiplication by $-1$ and is not a subspace. C is the solution set of a homogeneous linear system, which is a subspace. D is a plane that does not pass through the origin (the zero vector gives $4 \ne 0$), so it is not a subspace.