linear 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 dim V = 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.

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