Question

(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

Homework Answers

Answer #1

If any doubt pls comment

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
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}.
Let V and W be finite-dimensional vector spaces over F, and let φ : V →...
Let V and W be finite-dimensional vector spaces over F, and let φ : V → W be a linear transformation. Let dim(ker(φ)) = k, dim(V ) = n, and 0 < k < n. A basis of ker(φ), {v1, . . . , vk}, can be extended to a basis of V , {v1, . . . , vk, vk+1, . . . , vn}, for some vectors vk+1, . . . , vn ∈ V . Prove that...
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).
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.
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 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.
Suppose that V is a finite dimensional inner product space over C and dim V =...
Suppose that V is a finite dimensional inner product space over C and dim V = n, let T be a normal linear transformation of V If S is a linear transformation of V and T has n distinc eigenvalues such that ST=TS. Prove S is normal.
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 a linear map such that T2 - I = 0...
Let T: V -> V be a linear map such that T2 - I = 0 where I is the identity map on V. a) Prove that Im(T-I) is a subset of Ker(T+I) and Im(T+I) is a subset of Ker(T-I). b) Prove that V is the direct sum of Ker(T-I) and Ker(T+I). c) Suppose that V is finite dimensional. True or false there exists a basis B of V such that [T]B is a diagonal matrix. Justify your answer.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT