6. Let V be the vector space above. Consider the maps T : V → V...

6. Let V be the vector space above. Consider the maps T : V → V And S : V → V

defined by T(a1,a2,a3,...) = (a2,a3,a4,...) and S(a1,a2,a3,...) = (0,a1,a2,...).

(a) [optional] Show that T and S are linear.
(b) Show that T is surjective but not injective.
(c) Show that S is injective but not surjective.

(d) Show that V = im(T) + ker(T) but im(T) ∩ ker(T) ̸= {0}.

(e) Show that im(S) ∩ ker(S) = {0} but V ̸= im(S) + ker(S).

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

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.

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.

Let V be a vector space and let U and W be subspaces of V ....

Let V be a vector space and let U and W be subspaces of V . Show that the sum U + W = {u + w : u ∈ U and w ∈ W} is a subspace of V .

Let V be the vector space of all functions f : R → R. Consider the...

Let V be the vector space of all functions f : R → R. Consider the subspace W spanned by {sin(x), cos(x), e^x , e^−x}. The function T : W → W given by taking the derivative is a linear transformation a) B = {sin(x), cos(x), e^x , e^−x} is a basis for W. Find the matrix for T relative to B. b)Find all the eigenvalues of the matrix you found in the previous part and describe their eigenvectors. (One...

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.

Let T and S be linear transformations of a vector space V, and TS=ST (a) Show...

Let T and S be linear transformations of a vector space V, and TS=ST (a) Show that T preserves the generalized eigenspace and eigenspace of S. (b) Suppose V is a vector space on R and dimV = 4. S has a minimal polynomial of (t-2)2 (t-3)2?. What is the jordan canonical form of S. (c) Show that the characteristic polynomial of T has at most 2 distinct roots and splits completely.

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:...

Vector v=(9,0,2) is vector from R3 space. Consider standard inner product in R3. Let W be...

Vector v=(9,0,2) is vector from R3 space. Consider standard inner product in R3. Let W be a subspace in R3 span by u = (9,2,0) and w=(9/2,0,2). a) Does V belong to W? show explanation b) find orthonormal basis in W. Show work c) find projection of v onto W( he best approximation of v with elements of w) d) find the distance between projection and vector v

Let V be the vector space of 2 × 2 real matrices and let P2 be...

Let V be the vector space of 2 × 2 real matrices and let P2 be the vector space of polynomials of degree less than or equal to 2. Write down a linear transformation T : V ? P2 with rank 2. You do not need to prove that the function you write down is a linear transformation, but you may want to check this yourself. You do, however, need to prove that your transformation has rank 2.

Subjects