Suppose V is finite-dimensional and S, T are operators on V . Prove that ST is...

Suppose V is finite-dimensional and S, T are operators on V . Prove that ST is bijective if and only if S and T are both bijective.

Note: Don’t forget that bijective maps are precisely those that have an inverse!

Expert Solution

Colby Messinger answered 2 years ago

Let V be a finite-dimensional vector space over C and T in L(V). Prove that the...

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.

suppose that T : V → V is a linear map on a finite-dimensional vector space...

suppose that T : V → V is a linear map on a finite-dimensional vector space V such that dim range T = dim range T2. Show that V = range T ⊕null T. (Hint: Show that null T = null T2, null T ∩ range T = {0}, and apply the fundamental theorem of linear maps.)

Suppose U is a subspace of a finite dimensional vector space V. Prove there exists an...

Suppose U is a subspace of a finite dimensional vector space V. Prove there exists an operator T on V such that the null space of T is U.

Let T be a linear operator on a finite-dimensional complex vector space V . Prove that...

Let T be a linear operator on a finite-dimensional complex vector space V . Prove that T is diagonalizable if and only if for every λ ∈ C, we have N(T − λIV ) = N((T − λIV )2).

Let T be an operator on a finite-dimensional complex vector space V, and suppose that dim...

Let T be an operator on a finite-dimensional complex vector space V, and suppose that dim Null T = 3, dimNullT2 =6. Prove that T does not have a square root; i.e. there does not exist any S ∈ L (V) such that S2 = T.

Suppose ? is a finite-dimensional with dim ? > 1 and ? ∈ ℒ(?). Prove that...

Suppose ? is a finite-dimensional with dim ? > 1 and ? ∈ ℒ(?). Prove that {?(?)|? ∈ ?[?]} ≠ ℒ(?).

Show that if V is finite-dimensional and W is infinite-dimensional, then V and W are NOT...

Show that if V is finite-dimensional and W is infinite-dimensional, then V and W are NOT isomorphic.

Let V be a finite dimensional vector space over R. If S is a set of...

Let V be a finite dimensional vector space over R. If S is a set of elements in V such that Span(S) = V , what is the relationship between S and the basis of V ?

V and W are finite dimensional inner product spaces,T: V→W is a linear map 1A: Give...

V and W are finite dimensional inner product spaces,T: V→W is a linear map 1A: Give an example of a map T from R2 to itself (with the usual inner product) such that〈Tv,v〉= 0 for every map. 1B: Suppose that V is a complex space. Show that〈Tu,w〉=(1/4)(〈T(u+w),u+w〉−〈T(u−w),u−w〉)+(1/4)i(〈T(u+iw),u+iw〉−〈T(u−iw),u−iw〉 1C: Suppose T is a linear operator on a complex space such that〈Tv,v〉= 0 for all v. Show that T= 0 (i.e. that Tv=0 for all v).

6a. Let V be a finite dimensional space, and let Land T be two linear maps...

6a. Let V be a finite dimensional space, and let Land T be two linear maps on V. Show that LT and TL have the same eigenvalues. 6b. Show that the result from part A is not necessarily true if V is infinite dimensional.

Question