From 97ab2c89269162228ed81b1141b79853d595f09f Mon Sep 17 00:00:00 2001 From: Jeremy Yallop Date: Sat, 5 Aug 2023 13:34:41 +0100 Subject: [PATCH] Fix typos in definition of direct sum. --- ib/linalg/01_vector_spaces_and_linear_dependence.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/ib/linalg/01_vector_spaces_and_linear_dependence.tex b/ib/linalg/01_vector_spaces_and_linear_dependence.tex index 343fed34..8b578578 100644 --- a/ib/linalg/01_vector_spaces_and_linear_dependence.tex +++ b/ib/linalg/01_vector_spaces_and_linear_dependence.tex @@ -384,7 +384,7 @@ \subsection{Dimensionality of sums} \subsection{Direct sums} \begin{definition} Let \( V \) be an \( F \)-vector space and \( U, W \) be subspaces of \( V \). - We say that \( V = U \oplus V \), read as the direct sum of \( U \) and \( V \), if \( \forall v \in V, \exists! + We say that \( V = U \oplus W \), read as the direct sum of \( U \) and \( W \), if \( \forall v \in V, \exists! u \in U, \exists! w \in W, u + w = v \). We say that \( W \) is \textit{a} direct complement of \( U \) in \( V \); there is no uniqueness of such a complement.