Linear Algebra

188 Chapter Three. Maps Between Spaces(e) Show that the k-th derivative map is a linear transformation of P n for eachk. Prove that this map is a linear transformation of that spaced k−1f ↦→ dkdx f + c k k−1dx f + · · · + dk−1 c1dx f + c0ffor any scalars c k , . . . , c 0 . Draw a conclusion as above.2.41 Prove that for any transformation t: V → V that is rank one, the map givenby composing the operator with itself t ◦ t: V → V satisfies t ◦ t = r · t for somereal number r.2.42 Show that for any space V of dimension n, the dual spaceL(V, R) = {h: V → R ∣ ∣ h is linear}is isomorphic to R n . It is often denoted V ∗ . Conclude that V ∗ ∼ = V .2.43 Show that any linear map is the sum of maps of rank one.2.44 Is ‘is homomorphic to’ an equivalence relation? (Hint: the difficulty is todecide on an appropriate meaning for the quoted phrase.)2.45 Show that the rangespaces and nullspaces of powers of linear maps t: V → Vform descendingV ⊇ R(t) ⊇ R(t 2 ) ⊇ . . .and ascending{⃗0} ⊆ N (t) ⊆ N (t 2 ) ⊆ . . .chains. Also show that if k is such that R(t k ) = R(t k+1 ) then all followingrangespaces are equal: R(t k ) = R(t k+1 ) = R(t k+2 ) . . . . Similarly, if N (t k ) =N (t k+1 ) then N (t k ) = N (t k+1 ) = N (t k+2 ) = . . . .