Select all choices that are subspaces of R^3. Note: R denotes the set of real numbers.

≤ft\ x = bmatrix x_1 \\ x_2 \\ x_3 bmatrix ∈ R^3 : x = bmatrix 1 \\ 1 \\ 0 bmatrix + bmatrix 1 \\ 0 \\ 0 bmatrix, , ∈ R \

≤ft\ x = bmatrix x_1 \\ x_2 \\ x_3 bmatrix ∈ R^3 : 5x_1 + 2x_3 = 0,\ 4x_1 - 2x_2 + 3x_3 = 0 \

≤ft\ x = bmatrix x_1 \\ x_2 \\ x_3 bmatrix ∈ R^3 : x = bmatrix 1 \\ 1 \\ 0 bmatrix + bmatrix 1 \\ 0 \\ 0 bmatrix, , ∈ R \

≤ft\ x = bmatrix x_1 \\ x_2 \\ x_3 bmatrix ∈ R^3 : x = ^2 bmatrix 1 \\ 2 \\ 0 bmatrix + ^2 bmatrix 1 \\ 0 \\ 1 bmatrix, , ∈ R \

≤ft\ x = bmatrix x_1 \\ x_2 \\ x_3 bmatrix ∈ R^3 : 5x_1 + 2x_3 = 0,\ 4x_1 - 2x_2 + 3x_3 = 0 \

≤ft\ x = bmatrix x_1 \\ x_2 \\ x_3 bmatrix ∈ R^3 : 5x_1 + 2x_3 + 4 = 0 \

GATE DA 2024 Q47: Which sets are subspaces of R³

Linear Algebra
2 marks
MSQ
No negative marking

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

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

Try it: multiple select: one or more correct options

Select all correct options. No negative marking, no partial marks.

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Answer: A, C (official GATE 2024 final answer key)

Concept used: A subspace contains $\mathbf{0}$ and is closed under addition and scalar multiplication.

First step: Spans and solution sets of homogeneous equations are subspaces; $\alpha^2, \beta^2 \ge 0$ cannot give negative multiples, and an equation with $+4$ excludes $\mathbf{0}$ .

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Source: official GATE 2024 DA question paper and final answer key, organised by IISc Bangalore. Question paper · Answer key

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GATE DA 2024 Q47: Which sets are subspaces of R³

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