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 − λI_V ) = N((T − λI_V )²).

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Colby Messinger answered 2 years ago

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.

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.)

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.

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 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 ?

Let ? and W be finite dimensional vector spaces and let ?:?→? be a linear transformation....

Let ? and W be finite dimensional vector spaces and let ?:?→? be a linear transformation. We say a linear transformation ?:?→? is a left inverse of ? if ST=I_v, where ?_v denotes the identity transformation on ?. We say a linear transformation ?:?→? is a right inverse of ? if ??=?_w, where ?_w denotes the identity transformation on ?. Finally, we say a linear transformation ?:?→? is an inverse of ? if it is both a left and right...

1. Let ? be a finite dimensional vector space with basis {?1,...,??} and ? ∈ L(?)....

1. Let ? be a finite dimensional vector space with basis {?1,...,??} and ? ∈ L(?). Show the following are equivalent: (a) The matrix for ? is upper triangular. (b) ?(??) ∈ Span(?1,...,??) for all ?. (c) Span(?1,...,??) is ?-invariant for all ?. please also explain for (a)->(b) why are all the c coefficients 0 for all i>k? and why T(vk) in the span of (v1,.....,vk)? i need help understanding this.

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!

1. Let V be real vector space (possibly infinite-dimensional), S, T ∈ L(V ), and S...

1. Let V be real vector space (possibly infinite-dimensional), S, T ∈ L(V ), and S be in- vertible. Prove λ ∈ C is an eigenvalue of T if and only if λ is an eigenvalue of STS−1. Give a description of the set of eigenvectors of STS−1 associated to an eigenvalue λ in terms of the eigenvectors of T associated to λ. Show that there exist square matrices A, B that have the same eigenvalues, but aren’t similar. Hint:...

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