Exercise 1749Linear algebra › Inner product spacesL1Added 3 Nov 2026

Completing an orthonormal family

#orthonormal family#basis#completion

Statement

Let $(e_1,\ldots,e_k)$ be an orthonormal family in a finite-dimensional inner product space $E$ . Show that it can be completed to an orthonormal basis of $E$ .

Hint

Extend to a basis, then apply Gram-Schmidt while keeping the first $k$ vectors.

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Solution

Extension. By the incomplete basis theorem, extend $(e_1,\ldots,e_k)$ to a basis $(e_1,\ldots,e_k,v_{k+1},\ldots,v_n)$ of $E$ .

Gram-Schmidt, keeping the first $k$ . The vectors $e_1,\ldots,e_k$ are already orthonormal. Apply Gram-Schmidt to the whole basis: the first $k$ vectors don't change (they are already orthonormal), and the process produces orthonormal vectors $e_1,\ldots,e_k,e_{k+1},\ldots,e_n$ .

Conclusion. $(e_1,\ldots,e_n)$ is an orthonormal basis of $E$ completing the original family.

Completing an orthonormal family

Statement

Solution

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