Question

In: Advanced Math

1. Let T : Mn×n(F) → Mn×n(F) be the transposition map, T(A) = At. Compute the...

1. Let T : Mn×n(F) → Mn×n(F) be the transposition map, T(A) = At. Compute the characteristic polynomial of T. You may wish to use the basis of Mn×n(F) consisting of the matrices eij + eji, eij −eji and eii.

2.  Let A = (a b c d) (2 by 2 matrix) and let T : M2×2(F) → M2×2(F) be defined asT (B) = AB. Represent T as a 4×4 matrix using the ordered basis {e11,e21,e12,e22}, and use this matrix to prove that the characteristic polynomial of T is the square of the characteristic polynomial of A.

Solutions

Expert Solution


Related Solutions

Let F be a field and R = Mn(F) the ring of n×n matrices with entires...
Let F be a field and R = Mn(F) the ring of n×n matrices with entires in F. Prove that R has no two sided ideals except (0) and (1).
proof: L t^(n+1)*f(t)=(-1)^(n+1)*(d^(n+1)/ds^(n+1))*F(s)
proof: L t^(n+1)*f(t)=(-1)^(n+1)*(d^(n+1)/ds^(n+1))*F(s)
Let G be a group of order mn where gcd(m,n)=1 Let a and b be elements...
Let G be a group of order mn where gcd(m,n)=1 Let a and b be elements in G such that o(a)=m and 0(b)=n Prove that G is cyclic if and only if ab=ba
Let function F(n, m) outputs n if m = 0 and F(n, m − 1) +...
Let function F(n, m) outputs n if m = 0 and F(n, m − 1) + 1 otherwise. 1. Evaluate F(10, 6). 2. Write a recursion of the running time and solve it . 3. What does F(n, m) compute? Express it in terms of n and m.
Let A∈Mn(R)"> A ∈ M n ( R ) A∈Mn(R) such that I+A"> I + A I+A is invertible. Suppose that
Let A∈Mn(R)A∈Mn(R) such that I+AI+A is invertible. Suppose thatB=(I−A)(I+A)−1B=(I−A)(I+A)−1(a) Show that B=(I+A)−1(I−A)B=(I+A)−1(I−A) (b) Show that I+BI+B is invertible and express AA in terms of BB.
Let f(t) =t^2−1 and g(t) =e^t. (a) Graph f(g(t)) and g(f(t)). (b) Which is larger,f(g(5)) or...
Let f(t) =t^2−1 and g(t) =e^t. (a) Graph f(g(t)) and g(f(t)). (b) Which is larger,f(g(5)) or g(f(5))? Justify your answer. (c) Which is larger, (f(g(5)))′or g(f(5))′? Justify your answer.
Applied Math Let T be the operator on P2 defined by the equation T(f)=f+(1+x)f' (a) Show...
Applied Math Let T be the operator on P2 defined by the equation T(f)=f+(1+x)f' (a) Show T i linear operator from P2 into P2! (b) Give matrix reppressentaion in base vectorss B={1,x,x2}! (c) Give a diagonal matrix representing T (d) Give a diagonal matrix representing T Where P2 is ppolynomials with degree less then or equal to 2 and f' is the derivative of polynomial f.
Let f : N → N and g : N → N be the functions defined...
Let f : N → N and g : N → N be the functions defined as ∀k ∈ N f(k) = 2k and g(k) = (k/2 if k is even, (k + 1) /2 if k is odd). (1) Are the functions f and g injective? surjective? bijective? Justify your answers. (2) Give the expressions of the functions g ◦ f and f ◦ g? (3) Are the functions g ◦ f and f ◦ g injective? surjective? bijective?...
Let f(t)=5t2−t. a) Find f(t+h): b) Find f(t+h)−f(t): c) Find f(t+h)−f(t)/h: side note: (f(t+h)=f(t) is on...
Let f(t)=5t2−t. a) Find f(t+h): b) Find f(t+h)−f(t): c) Find f(t+h)−f(t)/h: side note: (f(t+h)=f(t) is on top of fraction and h is on bottom) d) Find f′(t): pls circle the 4 answers
Let f : Z × Z → Z be defined by f(n, m) = n −...
Let f : Z × Z → Z be defined by f(n, m) = n − m a. Is this function one to one? Prove your result. b. Is this function onto Z? Prove your result
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT