Exercise 597Linear algebra › Vector spaces and subspacesHigh schoolAdded 3 Nov 2026

Which subsets are subspaces?

#subspace criterion#counterexamples

Statement

In $\mathbb{R}^3$ , determine whether each subset is a subspace:

(a) $F_1 = \{(x,y,z) : x+y+z=0\}$ ;
(b) $F_2 = \{(x,y,z) : x+y+z=1\}$ ;
(c) $F_3 = \{(x,y,z) : x=y \text{ and } z=0\}$ ;
(d) $F_4 = \{(x,y,z) : x^2+y^2+z^2=0\}$ .

Hint

For each candidate, check whether $0$ belongs and whether the set is closed under addition and scalar multiplication.

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Solution

(a) Contains $0$ (since $0+0+0=0$ ). If $u=(x,y,z)$ and $v=(x',y',z')$ satisfy the equation, then $u+v$ satisfies it too, and $\lambda u$ also. So $F_1$ is a subspace (a plane through the origin).

(b) Does not contain $0$ (since $0+0+0=0\neq1$ ). So $F_2$ is not a subspace (it is an affine plane).

(c) Contains $0$ . If $u=(x,x,0)$ and $v=(x',x',0)$ , then $u+v=(x+x',x+x',0)\in F_3$ and $\lambda u=(\lambda x,\lambda x,0)\in F_3$ . So $F_3$ is a subspace (a line through the origin).

(d) $F_4=\{(0,0,0)\}$ : the only vector with norm $0$ is the zero vector. It is a subspace (the trivial subspace).

Conclusion. $F_1$ , $F_3$ , $F_4$ are subspaces; $F_2$ is not.

Which subsets are subspaces?

Statement

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