Question
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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