= Three projections summing to the identity
{title2=$E_1+E_2+E_3=I\Longrightarrow E_iE_j=0\quad(i\ne j)$}
Over a <field> of <characteristic> different from two, three <linear projections> summing to the identity annihilate one another in both orders. Each pair sum is the complementary projection to the third. The <sum of two linear projections> criterion therefore proves <zero> products. Every <vector> is the sum of its three images; applying each projection to a <zero> sum proves that these images form a <direct sum>. Choosing a <basis> in each image gives a simultaneous <eigenbasis>, with triples of <eigenvalues> $(1,0,0)$, $(0,1,0)$ or $(0,0,1)$. This is algebraic <orthogonality>, and does not assume that the projections are <orthogonal projections> for a specified <inner product>.
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