Definition 1.4.3.

  Let \(E\) be a \(K\)-vector space, and \(E_1\), \(E_2\) be two subspaces of \(E\). We say that \(E_1\) and \(E_2\) are supplementary if \(E = E_1 \oplus E_2\).

Example :

 \(E_1 = \{(x, 0) \mid x \in \mathbb{R}\} \subset \mathbb{R}^2\), \(E_2 = \{(0, y) \mid y \in \mathbb{R}\} \subset \mathbb{R}^2\), \(E_1 \oplus E_2 = \mathbb{R}^2\), \(E_1\) and \(E_2\) are supplementary. \), \(E_2 = \{(0, y) \mid y \in \mathbb{R}\} \subset \mathbb{R}^2\), \(E_1 \oplus E_2 = \mathbb{R}^2\), \(E_1\) and \(E_2\) are supplementary.

Proposition :

Let \(E_1\) and \(E_2\) be two finite-dimensional subspaces of a \(K\)-vector space \(E\) of arbitrary dimension. We have 

 1.\(\dim(E_1 \oplus E_2) = \dim E_1 + \dim E_2\).

 2.\(\dim(E_1 + E_2) = \dim E_1 + \dim E_2 - \dim(E_1 \cap E_2)\)