Prob le m 5 (Bonus 2 points) Let V be a finite dimensional vector space. Suppose that T : V -» V is matrix representati...
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.
QUESTION 5 Let V denote an arbitrary finite-dimensional vector space with dimension n E N Let B = {bi, bn} and B' = { bị, b, } denote two bases for V and let PB-B, be the transition matrix from B to B' Prove that where 1 V → V is the identity transformation, i e 1(v) v for all v E V Note that I s a linear transformation 14] QUESTION 5 Let V denote an arbitrary finite-dimensional vector...
3. Let V be a finite dimensional vector space with a positive definite scalar product. Let A: V-> V be a symmetric linear map. We say that A is positive definite if (Av, v) > 0 for all ve V and v 0. Prove: (a) if A is positive definite, then all eigenvalues are > 0. (b) If A is positive definite, then there exists a symmetric linear map B such that B2 = A and BA = AB. What...
Let V be a finite-dimensional complex vector space and let T from V to V be a linear transformation. Show that V is the direct sum of U and W where W and U are T-invariant subspaces and the restriction of T on U is nilpotent and the restriction of T on W is an isomorphism.
3. Let V be a finite dimensional inner product space, and suppose that T is a linear operator on this space. (i) Let B be an ordered orthonormal basis for V and let U be the linear operator on V determined by [U19 = (T);. Then, for all 01,09 € V, (01, T(02)) = (U(V1), v2) (ii) Prove that the conclusion of the previous part does not hold, in general, if the basis 8 is not orthonormal.
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
Prob 4. Let V be a finite-dimensional real vector space and let T є C(V). Define f : R R by f(A) :- dim range (T-λΓ Which condition on T is equivalent to f being a continuous function? Prob 4. Let V be a finite-dimensional real vector space and let T є C(V). Define f : R R by f(A) :- dim range (T-λΓ Which condition on T is equivalent to f being a continuous function?
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 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)
Let T be a linear operator on a finite dimensional vector space with a matrix representation A = 1 1 0 0] 16 3 2 1-3 -1 0 a. (3 pts) Find the characteristic polynomial for A. b. (3 pts) Find the eigenvalues of A. C. (2 pts) Find the dimension of each eigenspace of A. d. (2 pts) Using part (c), explain why the operator T is diagonalizable. e. (3 pts) Find a matrix P and diagonal matrix D...