Prob 4. Let V be a finite-dimensional real vector space and let T є C(V). Define f : R R by f(A) ...
Prob 4· Let V be a finite-dimensional vector space and let U be its proper subspace (i.e., UメV). Prove that there exists ф є V, 0 for all u є U but ф 0. such that p(u)
This is an Advanced Linear Algebra Question. Please answer only if your answer is fully sure. Do not copy answers from online!!! Do not copy answers from online!!! Prob 4. Let V be a finite-dimensional real vector space and let Te L(V). Define f : R R by f(A): dim range (T-AI) Which condition on T is equivalent to f being a continuous function? Hint: to be continuous f(A) is most likely to be a constant function since dimension would...
Prob 5. Suppose V is a finite-dimensional complex vector space, T E C(V) is diagonalizable, and all eigenvalues of T are strictly below 1 in absolute value. Given V, and u є V, what is limn oo (Tnt)?
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 :...
Problem 6. Let V be a vector space (a) Let (--) : V x V --> R be an inner product. Prove that (-, -) is a bilinear form on V. (b) Let B = (1, ... ,T,) be a basis of V. Prove that there exists a unique inner product on V making Borthonormal. (c) Let (V) be the set of all inner products on V. By part (a), J(V) C B(V). Is J(V) a vector subspace of B(V)?...
Problem #6. Let V be a finite dimensional vector space over a field F. Let W be a subspace of V. Define A(W) e Vw)Vw E W). Prove that A(W) is a subspace of (V).
Let V and W be finite dimensional vector spaces over R and T:V + W be linear. Let V be a subspace of V and Wo = T(V). (Select ALL that are TRUE) If T is surjective then Vo = {v EV : there is w E Wo such that T(v) = w} If T is injective then dim(VO) = dim(W). dim(ker(T) n Vo) = dim(VO) - dim(Wo).
• Let V be a 2-dimensional real vector space, and let T E End(V). Show that T is diagonalizable over C but not over R if and only if tr(T)2 < 4. det(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 ,
Let V be a finite-dimensional vector space, and let f :V + V be a linear map. Let also A be a matrix representation of f in some basis of V. As you know, any other matrix representation of f is similar to A. Show, conversely, that every matrix similar to A is a matrix representation of f with respect to some basis of V.