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Eigenspaces and the Fundamental Theorem of Algebra

Norbert A'Campo

The following well-known theorem will be proved first, and then the so-called F undamental Theorem of Algebra will be deduced from it.

Theorem 1   Let $V$ be a ${\bf C}$-vectorspace of finite dimension, $\mathop{\rm dim}V\geq 2$, and let $A:V\to V$ be a linear transformation. Then there exists a linear subspace $W$ in $V$ with $A(W)\subset W $ and $ 0< \mathop{\rm dim}W < \mathop{\rm dim}V $.

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