Let $V$ be a vector space and $X$, $Y\subseteq V$ be subspaces. The sum $X+Y$ is direct if $X\cap Y=\{0\}$. The notation
\[W = X \oplus Y\]means
\[W = X + Y \enspace\land\enspace X\cap Y = \{0\}.\]If $V=X\oplus Y$, $X$ and $Y$ are called complementary subspaces to each other.
Let $V$ be a vector space and $X$, $Y\subseteq V$ be subspaces such that $V=X+Y$. The following statements are equivalent:
- $V = X\oplus Y$.
- For each $v\in V$ there are unique $x\in X$ and $y\in Y$ such that $v=x+y$.
- For any $x\in X$ and $y\in Y$, if $x\neq 0$ and $y\neq 0$, then $x$ and $y$ are linear independent.
Let $V$ be a vector space and $X$, $Y\subseteq V$ be finite-dimensional subspaces. The following statements are equivalent:
- $V = X \oplus Y$.
- $X$ has a basis $\mathcal{A}$ and $Y$ has a basis $\mathcal{B}$ such that $\mathcal{A}\cup\mathcal{B}$ is a basis of $V$.
- $V = X + Y$ and $\dim V = \dim X + \dim Y$.
Notation
- $\dim V$
- The dimension of $V$.