Let V be a finite dimensional vector space and TE L(VV), such that, T30. a) Show...
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...
3. Let Te L(V), where V is a finite-dimensional C-vector space. Prove that T is diago- nalizable if and only if Ker(T – a id) n Im(T - a id) = {0} for all a E C.
Let V be a finite-dimensional vector space, and let B be a basis of V. Show that there is an inner product on V for which B is orthonormal.
Let V be a finite dimensional inner product space, w1,w2V. Let TL(V) and Tv=<v,w1>w2 for all vV. Find all eigenvalues and the corresponding eigenspaces of T. Please provide full solution. We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Question 1. Let V be a finite dimensional vector space over a field F and let W be a subspace of Prove that the quotient space V/W is finite dimensional and dimr(V/IV) = dimF(V) _ dimF(W). Hint l. Start with a basis A = {wi, . . . , w,n} for W and extend it to a basis B = {wi , . . . , wm, V1 , . . . , va) for V. Hint 2. Our goal...
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.
Q-) Let F be an object ond V is a finite dimensional vector Space on the object. . that if v is linear trons formation, ronkt is zero a) Show or 1. b) If Liv> v is linear tronsformation, show that ker L c ker L² and L(v) 2 L² (v). ( Note : L²=LoL and ker L, be defined as subspace of L.).
Let V be a finite-dimensional vector space over C and T in L(V). Prove that the set of zeros of the minimal polynomial of T is exactly the same as the set of the eigenvalues of T.
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 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.