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Which one of the following is a subspace of the vector space ℝ3?

Which one of the following is a subspace of the vector space ℝ3? Correct Answer {(x, y, z) ϵ ℝ<sup>3</sup> : x + 2y = 0, 2x + 3z = 0}

Let v1 = {(x, y, z)|x + 2y = 0, 2x + 3z = 0}

Let u1 {x1, y, z} & u2 = (x2, y2, z2) ϵ V1

u1 + u2 = (x1 + x2, y1 + y2, z1 + z2) such that (x1 + x2) + 2(y1 + y2) = (x1 + 2y1) + (x2 + 2y2)

= 0 + 0 = 0

u1 + u2 ϵ V1

Next, α u = α (x, y, z) = (α x, α y, α z) such that (α x + 2 α y) = α (x + 2y) = 0

2(α x) + 3(α z) = α (2x + 3z) = 0

⇒ α u ϵ v1

∴  V1 is a subspace of vector space R3

Further V2 {(x, y, z)|2x + 3y + 4z = 3, z = 0}

If u1 & u2 ϵ V2, then

u1 + u2 = (x1 + x2, y1 + y2, z1 + z2) such that 2(x1 + x2) + 3(y1 + y2) + 4(z1 + z2) = 3

 (2x1 + 3y1 + 4z1) + (2x2 + 3y2 + 4z2) = 3

⇒ 3 + 3 ≠ 3

⇒ V2 is not a subspace.

Similarly we can show that (c) & (b) does not satisfy subspace conditions.

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