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In my book is stated (in Dutch, but I tried to translate it to English):

Let $V$ be a $K$-vectorspace and $q_1, \dots, q_r$ a set of projection-operators on $V : \sum^r_{i=1}{q_i} = 1_V$ and $\forall i \neq j, q_i q_j = 0$. Let $V_i =$ im $q_i$. Then $V = V_1 \oplus \dots \oplus V_r$

I am confused by the notation $1_V$, what does it mean?

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up vote 2 down vote accepted

I suspect it means the identity function $1= \mathrm{id}_V: V \to V$. It looks as if $\sum^r_{i=1}{q_i} = 1_V = q_1 \oplus \dots \oplus q_r$ which is the identity map since $q_i : V_i \to V_i$ is the identity map.

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