Epub ISBNs: 9781118496701 ; 9781118496732
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Additional resources for Calculus: 1,001 Practice Problems for Dummies (+ Free Online Practice)
6 Let U and V be finite dimensional vector spaces and τ ∈ L(U, V ). Then the map τ ∗ : φ → φ ◦ τ belongs to L(V ∗ , U ∗ ) and induces an isomorphism ∗ : L(U, V ) ∼ = L(V ∗ , U ∗ ). 7 With the above notation, τ ∗ is called the dual of τ . Note that if σ ∈ L(V, W ), then (σ ◦ τ )∗ (ψ) = ψ ◦ (σ ◦ τ ) = τ ∗ (ψ ◦ σ) = τ ∗ (σ ∗ (ψ)) for all ψ ∈ W ∗ . That is, (στ )∗ = τ ∗ σ ∗ . 8 Let U and V be finite dimensional vector spaces with bases A and B and let τ ∈ L(U, V ). If τ is represented by T with respect to these bases, then τ ∗ is represented by T t with respect to the dual bases.
Consider the orthogonal matrix cosθ −sinθ sinθ cosθ Viewed as a matrix over R, there are no eigenvalues if θ ∈ π Z since each vector v = 0 is rotated by θ and so does not belong to the subspace generated by v. Hence the matrix is not diagonalisable over R. However, over C, there are two eigenvalues eiθ and e−iθ and the matrix is diagonalisable (do this). Note that the minimal polynomial over R is X 2 − 2Xcosθ + 1 and this has no roots in R; over C, the minimal polynomial is the same and can be written as (X − eiθ )(X − e−iθ ).
0 0 0 0 0 0 1 0 ... ... 0 ... µj 0 0 ... 0 0 .. 1 µj Then J (d1 ) (µ1 ) ⊕ ... ⊕ J (d ) (µ ) is the Jordan form for τ = DV . 4 -1. Exercises Show that none of the following matrices are conjugate: 1 1 0 0 1 1 0 0 1 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 48 CHAPTER 3. JORDAN NORMAL FORM Is the matrix 1 1 1 A= 0 1 1 0 0 1 conjugate to any of them? If so, which? ] 2. Show that any n × n matrix over C with all its eigenvalues real is conjugate to an n × n matrix over R.