Sum of Vector Spaces

VSUM#DEF. Sum of Vector Spaces.

Let $V$ be a vector space and $X$, $Y\subseteq V$ be subspaces. The sum of $X$ and $Y$ is

\[X + Y = \{x+y : x\in X, y\in Y\}.\]

VSUM#PROP-S.

Let $V$ be a vector space. For any subspaces $X$, $Y\subseteq V$:

1. $X+Y$ is a subspace.
2. $X+Y = \langle X\cup Y\rangle$.

VSUM#PROP-DIM.

Let $V$ be a vector space. For any finite-dimensional subspaces $X$, $Y\subseteq V$,

\[\dim(X+Y) = \dim X + \dim Y - \dim(X\cap Y).\]

As a corollary:

VSUM#PROP-DIM-I.

Let $V$ be a vector space. For any finite-dimensional subspaces $X$, $Y\subseteq V$,

\[\dim(X+Y) \leq \dim X + \dim Y.\]

Notation

$\dim V$

The dimension of $V$.