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1. Let ??(?) = ∑︀??=1 ??? denote the operation of taking the trace of ? ∈...

1. Let ??(?) = ∑︀??=1 ??? denote the operation of taking the trace of ? ∈ ??(F). Show that (by constructing a proof) ??(??+??) = ???(?)+???(?) for any ?,? ∈ ??(F) and ?,? ∈ F. Conclude that ?? is a linear transformation from ??(F) to F.

2. For a fixed ? ∈ R, determine the dimension of the subspace of P?(R) defined by {? ∈ P?(R) | ?(?) = 0}.

3. Let ? be a finite dimensional vector space. Suppose that ? ∈ L(?, ? ) is one to one and {?1,...,??} is a linearly independent subset of ?. Show that {?(?1),...,?(??)} is linearly independent in W.

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