Semilinear/Composition/Induction/Exercise

Let $V_1,\dots ,V_n,V_{n+1}$ be vector spaces over $ℂ$, and let

${\phi }_i:V_i⟶V_{i+1}$

denote linear or antilinear mappings, and let

$\phi ={\phi }_n\circ {\phi }_{n-1}\circ \cdots \circ {\phi }_2\circ {\phi }_1$

denote the composition of the mappings. Show by induction over $n$ the following two statements.

a) If the number of the antilinear mappings is even, then $\phi$ is linear.

b) If the number of the antilinear mappings is odd, then $\phi$ is antilinear.

Does the converse of this statement hold?

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