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Endomorphism/Simple properties of invariant subspaces/Exercise

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Let a linear mapping on a -vector space over a field . Show the following properties.

  1. The zero space is -invariant.
  2. is -invariant.
  3. Eigenspaces are -invariant.
  4. Let be -invariant linear subspaces. Then also and are -invariant.
  5. Let be a -invariant linear subspace. Then also the image space and the preimage space are -invariant.