Direct Sum of Vector Spaces

Definition 1. 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.

Proposition 1. Let $V$ be a vector space and $X$, $Y \subseteq V$ be subspaces such that $V = X + Y$. The following statements are equivalent:

1. \[V = X \oplus Y.\]

2. For each $v \in V$ there are unique $x \in X$ and $y \in Y$ such that $v = x + y$.

3. For any $x \in X$ and $y \in Y$, if $x \neq 0$ and $y \neq 0$, then $x$ and $y$ are linear independent.

Proposition 2. Let $V$ be a vector space and $X$, $Y \subseteq V$ be finite-dimensional subspaces. The following statements are equivalent:

1. \[V = X \oplus Y.\]
2. \[V = X + Y \enspace\land\enspace \dim V = \dim X + \dim Y.\]
3. $X$ has a basis $\mathcal{A}$ and $Y$ has a basis $\mathcal{B}$ such that $\mathcal{A} \cup \mathcal{B}$ is a basis of $V$.