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1. Suppose that ? is a finite dimensional vector space over R. Show that if ???(?...

1. Suppose that ? is a finite dimensional vector space over R. Show that if ???(? ) is odd, then every ? ∈ L(? ) has an eigenvalue. (Hint: use induction).

(please provide a detailed proof)

2. Suppose that ? is a finite dimensional vector space over R and ? ∈ L(? ) has no eigenvalues. Prove that every ? -invariant subspace of ? has even dimension.
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