Advanced Linear Algebra
《Advanced Linear Algebra》 · 第 41 页 · Chapter 1「Direct Sums」
Definition A vector space $V$ is the (internal) direct sum of a family $\mathcal S=\{S_i\mid i\in I\}$ of subspaces of $V$ if $V=\sum_{i\in I}S_i$ and $S_i\cap\sum_{j\ne i}S_j=\{0\}$ for each $i\in I$. Equivalently, every nonzero $v\in V$ has a unique, except for order of terms, expression as a sum of nonzero vectors from distinct subspaces in $\mathcal S$.