Question

Let V be a finite-dimensional vector space over C and T in L(V) be an invertible...

Let V be a finite-dimensional vector space over C and T in L(V) be an invertible operator in V. Suppose also that T=SR is the polar decomposition of T where S is the correspondIng isometry and R=(T*T)^1/2 is the unique positive square root of T*T. Prove that R is an invertible operator that committees with T, that is TR-RT.

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let V be a finite-dimensional vector space and let T be a linear map in L(V,...
Let V be a finite-dimensional vector space and let T be a linear map in L(V, V ). Suppose that dim(range(T 2 )) = dim(range(T)). Prove that the range and null space of T have only the zero vector in common
Let V be a vector space: d) Suppose that V is finite-dimensional, and let S be...
Let V be a vector space: d) Suppose that V is finite-dimensional, and let S be a set of inner products on V that is (when viewed as a subset of B(V)) linearly independent. Prove that S must be finite e) Exhibit an infinite linearly independent set of inner products on R(x), the vector space of all polynomials with real coefficients.
(3) Let V be a finite dimensional vector space, and let T: V® V be a...
(3) Let V be a finite dimensional vector space, and let T: V® V be a linear transformation such that rk(T) = rk(T2). a) Show that ker(T) = ker(T2). b) Show that 0 is the only vector that lies in both the null space of T, and the range space of T
Let V be a finite dimensional vector space over R with an inner product 〈x, y〉...
Let V be a finite dimensional vector space over R with an inner product 〈x, y〉 ∈ R for x, y ∈ V . (a) (3points) Let λ∈R with λ>0. Show that 〈x,y〉′ = λ〈x,y〉, for x,y ∈ V, (b) (2 points) Let T : V → V be a linear operator, such that 〈T(x),T(y)〉 = 〈x,y〉, for all x,y ∈ V. Show that T is one-to-one. (c) (2 points) Recall that the norm of a vector x ∈ V...
Let T:V→V be an endomorphism of a finite dimensional vector space over the field Z/pZ with...
Let T:V→V be an endomorphism of a finite dimensional vector space over the field Z/pZ with p elements, satisfying the equation Tp=T. Show that T is diagonalisable.
Let V be a finite dimensional vector space and T ∈ L(V : V ), such...
Let V be a finite dimensional vector space and T ∈ L(V : V ), such that, T3 = 0. a) Show that the spectrum of T is σ(T) = {0}. b) Show that T cannot be diagonalized (unless we are in the trivial case T = O).
1. Suppose that ? is a finite dimensional vector space over R. Show that if ???(?...
1. Suppose that ? is a finite dimensional vector space over R. Show that if ???(? ) is odd, then every ? ∈ L(? ) has an eigenvalue. (Hint: use induction). (please provide a detailed proof) 2. Suppose that ? is a finite dimensional vector space over R and ? ∈ L(? ) has no eigenvalues. Prove that every ? -invariant subspace of ? has even dimension.
Let V be a finite dimensional space over R, with a positive definite scalar product. Let...
Let V be a finite dimensional space over R, with a positive definite scalar product. Let P : V → V be a linear map such that P P = P . Assume that P T P = P P T . Show that P = P T .
Let V be an n-dimensional vector space and W a vector space that is isomorphic to...
Let V be an n-dimensional vector space and W a vector space that is isomorphic to V. Prove that W is also n-dimensional. Give a clear, detailed, step-by-step argument using the definitions of "dimension" and "isomorphic" the Definiton of isomorphic:  Let V be an n-dimensional vector space and W a vector space that is isomorphic to V. Prove that W is also n-dimensional. Give a clear, detailed, step-by-step argument using the definitions of "dimension" and "isomorphic" The Definition of dimenion: the...
) Let L : V → W be a linear transformation between two finite dimensional vector...
) Let L : V → W be a linear transformation between two finite dimensional vector spaces. Assume that dim(V) = dim(W). Prove that the following statements are equivalent. a) L is one-to-one. b) L is onto. please help asap. my final is tomorrow morning. Thanks!!!!
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT