(2) Suppose that W is finite dimensional and T E (V, W). Prove that T is injective if and only if...
(1) Suppose that V and W are both finite dimensional vector spaces. Prove that there exists a surjective linear map from V onto W if and only if Dim(W) Dim(V)
Prove the following → V such that (a) If T:V + W is linear and injective, then there exists a linear map S: W ST = I. (b) If S: W → V is linear and surjective, then there exists a linear map T:V ST = 1. W such that
Let V be a finite-dimensional vector space and let T L(V) be an operator. In this problem you show that there is a nonzero polynomial such that p(T) = 0. (a) What is 0 in this context? A polynomial? A linear map? An element of V? (b) Define by . Prove that is a linear map. (c) Prove that if where V is infinite-dimensional and W is finite-dimensional, then S cannot be injective. (d) Use the preceding parts to prove...
de(V)e(A) 6· Lot V be a finite dimensional vector space, and T : V → V be a linear oper- ator. Suppose that T2-Iv, the identity operator. Prove that T is diagonalizable and that 1 and1 are the only possible eigenvalues of T
Problem 6. Let V, W, and U be finite-dimensional vector spaces, and let T : V → W and S : W → U be linear transformations (a) Prove that if B-(Un . . . , v. . . . ,6) is a basis of V such that Bo-(Un .. . ,%) s a basis of ker(T) then (T(Fk+), , T(n)) is a basis of im(T) (b) Prove that if (w!, . . . ,u-, υ, . . . ,i)...
Vectors pure and applied Exercise 11.5.9 Let U and V be finite dimensional spaces over F and let θ : U linear map. v be a (i) Show that o is injective if and only if, given any finite dimensional vector space W map : V W such that over F and given any linear map α : U-+ W, there is a linear (ii) Show that θ is surjective if and only if, given any finite dimensional vector space...
Exercise 12.6.3 Let V and W be finite dimensional vector spaces over F, let U be a subspace of V and let α : V-+ W be a surjective linear map, which of the following statements are true and which may be false? Give proofs or counterexamples O W such that β(v)-α(v) if v E U, and β(v) (i) There exists a linear map β : V- otherwise (ii) There exists a linear map γ : W-> V such that...
8 Suppose V is finite-dimensional and P E L(V) is such that P2 = P. and || P v|| = || V || for every v E V. Prove that there exists a subspace U of V such that P = Pu.
Prob le m 5 (Bonus 2 points) Let V be a finite dimensional vector space. Suppose that T : V -» V is matrix representation with respect to every basis of V. Prove that the dimension of linear transform ation that has the same that T must be a scalar multiple of the identity transformation. You can assume V is 3 Prob le m 5 (Bonus 2 points) Let V be a finite dimensional vector space. Suppose that T :...
span(p) for some pe W, Prob 2. Suppose V and Ware finite-dimensional, T є L(VW ), and null T' Prove that range T-null 4. Give an example of such a pair T 0 メ0 for V-IR2, W-R3 ,