Exercise 1928Linear algebra › Vector spaces and subspacesL1Added 3 Nov 2026

Sum of two planes in $\mathbb{R}^3$

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Statement

Let $F=\{x+y+z=0\}$ and $G=\{x-y=0\}$ in $\mathbb{R}^3$ . Compute $F\cap G$ , $F+G$ , and determine whether $F$ and $G$ are supplementary.

Hint

Compute the intersection first, then apply $\dim(F+G)=\dim F+\dim G-\dim(F\cap G)$ .

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Solution

Intersection. Solve $x+y+z=0$ and $x=y$ : $2x+z=0$ , so $z=-2x$ . Thus $F\cap G=\operatorname{span}((1,1,-2))$ , dimension $1$ .

Sum. $\dim(F+G)=2+2-1=3$ , so $F+G=\mathbb{R}^3$ .

Supplementary? For $F$ and $G$ to be supplementary, we need $F\cap G=\{0\}$ , but here the intersection has dimension $1$ . So they are not supplementary.

Conclusion. $F\cap G$ is a line, $F+G=\mathbb{R}^3$ , but $F\oplus G\neq\mathbb{R}^3$ (they are not in direct sum).

Sum of two planes in $\mathbb{R}^3$

Statement

Solution

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