 Common plane for three pairwise complementary subspaces

ID: common-plane-for-three-pairwise-complementary-subspaces

V = A_{1} \oplus A_{2} = A_{2} \oplus A_{3} = A_{1} \oplus A_{3}

with nonzero subspaces, choose

0 \neq = v \in A_{3}

and write

v = a_{1} + a_{2}

using the first direct sum of vector spaces. Both components are nonzero. The plane spanned by them meets

A_{1}

and

A_{2}

in their component lines and meets

A_{3}

in the line through

v

. Its intersection with

A_{3}

cannot be the whole plane, since

a_{1} \in / A_{3}

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