Exercise 1222Linear algebra › Vector spaces and subspacesL1Added 3 Nov 2026

Subspace inclusion and dimension

#inclusion#dimension#equality

Statement

Let $F$ and $G$ be subspaces of a finite-dimensional space $E$ with $F\subseteq G$ . Show that if $\dim F=\dim G$ , then $F=G$ .

Hint

Take a basis of $F$ ; it is a family of $\dim F$ independent vectors in $G$ . If $\dim G=\dim F$ , it is also a basis of $G$ .

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Solution

Argument. Let $(e_1,\ldots,e_k)$ be a basis of $F$ , where $k=\dim F$ . Since $F\subseteq G$ , these vectors belong to $G$ and are independent. Since $\dim G=k$ , any independent family of $k$ vectors in $G$ is a basis of $G$ . Hence $(e_1,\ldots,e_k)$ is a basis of $G$ , so $G=\operatorname{span}(e_1,\ldots,e_k)=F$ .

Conclusion. $F=G$ . This is a standard shortcut: inclusion + equal dimension = equality.

Subspace inclusion and dimension

Statement

Solution

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