Question

Suppose V is a vector space and T is a linear operator. Prove by induction that...

Suppose V is a vector space and T is a linear operator. Prove by induction that for all natural numbers n, if c is an eigenvalue of T then c^n is an eigenvalue of T^n.
Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
1. Assume that V is a vector space and L is a linear function V →...
1. Assume that V is a vector space and L is a linear function V → V. a. Suppose there are two vectors v and w in V such that v, w, and v+w are all eigenvectors of L. Show that v and w share the same eigenvalue. b. Suppose that every vector in V is an eigenvector of L. Prove that there is a scalar α such that L = αI.
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 T be a 1-1 linear transformation from a vector space V to a vector space...
Let T be a 1-1 linear transformation from a vector space V to a vector space W. If the vectors u, v and w are linearly independent in V, prove that T(u), T(v), T(w) are linearly independent in W
Suppose that V is a vector space with basis {u, v, w}. Suppose that T is...
Suppose that V is a vector space with basis {u, v, w}. Suppose that T is a linear transformation from V to W and suppose also that {T(u), T(v), T(w)} is a basis for W. Prove from the definitions that T is both 1-1 and onto.
Suppose V is a vector space over F, dim V = n, let T be a...
Suppose V is a vector space over F, dim V = n, let T be a linear transformation on V. 1. If T has an irreducible characterisctic polynomial over F, prove that {0} and V are the only T-invariant subspaces of V. 2. If the characteristic polynomial of T = g(t) h(t) for some polynomials g(t) and h(t) of degree < n , prove that V has a T-invariant subspace W such that 0 < dim W < n
Suppose V is a finite dimensional inner product space. Prove that every orthogonal operator on V...
Suppose V is a finite dimensional inner product space. Prove that every orthogonal operator on V , i.e. <T(u), T(v)> , ∀u,v ∈ V , is an isomorphism.
4. Prove the Following: a. Prove that if V is a vector space with subspace W...
4. Prove the Following: a. Prove that if V is a vector space with subspace W ⊂ V, and if U ⊂ W is a subspace of the vector space W, then U is also a subspace of V b. Given span of a finite collection of vectors {v1, . . . , vn} ⊂ V as follows: Span(v1, . . . , vn) := {a1v1 + · · · + anvn : ai are scalars in the scalar field}...
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.
† Let β={v1,v2,…,vn} be a basis for a vector space V and T:V→V be a linear...
† Let β={v1,v2,…,vn} be a basis for a vector space V and T:V→V be a linear transformation. Prove that [T]β is upper triangular if and only if T(vj)∈span({v1,v2,…,vj}) j=1,2,…,n. Visit goo.gl/k9ZrQb for a solution.
5. Let V be a finite-dimension vector space and T : V → V be linear....
5. Let V be a finite-dimension vector space and T : V → V be linear. Show that V = im(T) + ker(T) if and only if im(T) ∩ ker(T) = {0}.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT