GATE DA 2024 Set 1 — Question 47
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Linear Algebra → Vector Spaces, Basis & Dimensions → Subspaces
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Question
Select all choices that are subspaces of .Note: denotes the set of real numbers.
Correct answer
(A) \ x = bmatrix x₁ \ x₂ \ x₃ bmatrix ∈ R³: x = α bmatrix 1 \ 1 \ 0 bmatrix + β bmatrix 1 \ 0 \ 0 bmatrix, α, β ∈ R \; (C) \ x = bmatrix x₁ \ x₂ \ x₃ bmatrix ∈ R³: 5x₁ + 2x₃ = 0, 4x₁ - 2x₂ + 3x₃ = 0 \
Solution
A subset is a subspace if it contains the zero vector and is closed under vector addition and scalar multiplication.(A) This set is the span of two vectors. The span of any set of vectors is a subspace. True.(B) The coefficients are and , which are non-negative. This set is a cone, not a subspace. For example, if vector is in the set (with ), is not in the set because coefficients cannot be negative. False.(C) This is the solution set of a homogeneous system of linear equations (). The solution space (null space) is always a subspace. True.(D) This is the solution set of a non-homogeneous equation (). It does not contain the zero vector. False.
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