Exercise 1223Linear algebra › Vector spaces and subspacesL1Added 3 Nov 2026

The direct sum criterion

#direct sum#criterion#proof

Statement

Let $F$ and $G$ be subspaces of a finite-dimensional vector space $E$ . Show that $F+G$ is a direct sum if and only if $\dim(F+G)=\dim F+\dim G$ .

Hint

Use the dimension formula $\dim(F+G)=\dim F+\dim G-\dim(F\cap G)$ .

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Solution

If $F+G$ is direct. Then $F\cap G=\{0\}$ , so $\dim(F\cap G)=0$ . The dimension formula gives $\dim(F+G)=\dim F+\dim G$ .

Conversely. If $\dim(F+G)=\dim F+\dim G$ , the dimension formula gives $\dim(F\cap G)=0$ , so $F\cap G=\{0\}$ . This is exactly the definition of a direct sum.

Conclusion. $F+G$ is direct if and only if the dimensions add.

The direct sum criterion

Statement

Solution

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