Do the solutions of x + y + z = 0 form a subspace of ℝ³? And those of x + y + z = 6?

Both are subspaces: each is a plane

Neither: subspaces must be lines

Physics I / Vector Spaces and Subspaces

Practice question · Multiple choice

Do the solutions of $x + y + z = 0$ form a subspace of $\mathbb{R}^{3}$ ? And those of $x + y + z = 6$ ?

Hints

Run the checklist on each: does it contain $\mathbf{0}$ ? Is it closed under $+$ and scalar $\times$ ?
Does $(0,0,0)$ satisfy $x+y+z=6$ ? And if $u, v$ both sum to 6, what does $u + v$ sum to?

Show the answer

B. First yes, second no

Why

The plane through the origin passes every test; the shifted plane fails immediately ( $\mathbf{0}$ missing, sums drift to 12). Subspaces must pass through the origin, homogeneous equations ( $= 0$ ) build subspaces, inhomogeneous ones ( $= c$ ) build offset copies that are not.

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Do the solutions of x+y+z=0x + y + z = 0x+y+z=0 form a subspace of R3\mathbb{R}^{3}R3? And those of x+y+z=6x + y + z = 6x+y+z=6?

Practise Vector Spaces and Subspaces

More questions on Vector Spaces and Subspaces

Do the solutions of $x + y + z = 0$ form a subspace of $\mathbb{R}^{3}$ ? And those of $x + y + z = 6$ ?