1. Let V be a vector space with bases B and C. Suppose that T:V V is a linear map with matrix representations Ms(T)A and Me(T) B. Prove the following (a) T is one-to-one iff A is one-to-one. (b) λ is an eigenvalue of T iff λ is an eigenvalue of B. Consequently, A and B have the same eigenvalues (c) There exists an invertible matrix V such that A-V-BV 1. Let V be a vector space with bases B...
Prove the following → V such that (a) If T:V + W is linear and injective, then there exists a linear map S: W ST = I. (b) If S: W → V is linear and surjective, then there exists a linear map T:V ST = 1. W such that
Let V and W be finite dimensional vector spaces and let T:V → W be a linear transformation. We say a linear transformation S :W → V is a left inverse of T if ST = Iy, where Iy denotes the identity transformation on V. We say a linear transformation S:W → V is a right inverse of T if TS = Iw, where Iw denotes the identity transformation on W. Finally, we say a linear transformation S:W → V...
Suppose V is finite-dimensional, T:V V is a linear operator, and (T-21)(T-31)(T-41) = 0. Show (without resorting to the Cayley-Hamilton theorem) that if is an eigenvalue of T, then = 2, 3, or 4. Suggestion: compute (T – 21)(T – 31)(T – 41)Ū, where ū is an eigen vector of Twith eigenvalue .
3. If T and S are similar in L(V) and also invertible does that imply that their inverses are similar?
Let W be a subspace of an n-dimensional vector space V over C, and let T:V V be a linear transformation. Prove that W is invariant under T if and only if W is invariant under T- I for any i EC.
8. Suppose V is an n-dimensional complex vector space. Suppose T E C(V) is such that 1,2, and 3 are the only distinct eigenvalues of T (a) Prove that the dimension of each generalized eigenspace of T is at most (n - 2). (b) Show that (T-1)"-2(T-21)"-"(7-31)"-"(a) = 0V, for all α є V. 8. Suppose V is an n-dimensional complex vector space. Suppose T E C(V) is such that 1,2, and 3 are the only distinct eigenvalues of T...
could somone plz help with #4 3. If T : p → W is a linear transformation, then T is one-to-one if and only if ker T = {0} 4. Prove that if T:V-is a linear transformation and W is a subspace of V, then the image of W'is a subspace of V" 3. If T : p → W is a linear transformation, then T is one-to-one if and only if ker T = {0} 4. Prove that if...
Suppose T: V V is a linear operator. Suppose p(x) = (1-r)(x- s) has distinct real roots (rs) and that and p(T) is the zero operator. Show that V is spanned by eigenvectors of T with eigenvalues r and s. Suppose T: V V is a linear operator. Suppose p(x) = (1-r)(x- s) has distinct real roots (rs) and that and p(T) is the zero operator. Show that V is spanned by eigenvectors of T with eigenvalues r and s.
Let V and W be two vector spaces over R and T:V + W be a linear transformation. We call a linear map S: W → V a generalized inverse of T if To SOT = T and SoTo S = S. If V and W are finite dimensional, show that there exists a generalized inverse of T.