28 May inear algebra
MA 351 HOMEWORK 4
Due: 7th March, 2018
Question 1 (5 points) Let σ : V → W be an isomorphism with dim(V ) = n. Define a map σ−1 : W → V by setting
σ−1(w) = v where v ∈ V is a vector such that σ(v) = w.
Show that σ−1 is a well-defined isomorphism from W to V . Moreover, the composite
σ−1 ◦ σ = idV , the identity map on V , and σ ◦ σ−1 = idW .
Question 2 (8 points) Let T : V → V be a morphism on V with dimV = n. Pick an isomorphism σ : V → Rn and one obtains a morphism
Tσ := σ ◦ T ◦ σ−1 : Rn → Rn.
Let Aσ be the unique n × n matrix representing Tσ, i.e., Tσ(x) = Aσx for all x ∈ Rn. Prove:
(i) For any two choices of isomorphisms σ, τ : V → Rn, there exists an n×n invertible matrix P such that
Aτ = PAσP −1.
(Hint: consider τ ◦ σ−1.) (ii) Define the characteristic polynomial det(λ · idV − T ) of T to be
det(λ · idV − T ) := det(λIn − Aσ)
by choosing an isomorphism σ : V → Rn. Prove that it is well-defined, indepen- dent of the chosen σ.
Question 3 (7 points). Let
A =
1 1 00 1 0 0 0 1
, B = 1 1 01 1 0
0 0 1
. Are they diagonalisable? If not, justify your answer; if yes, find the corresponding ma-
trices P and D.
1
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