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(1 point) Let V be a vector space, and T:V → V a linear transformation such...
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.
Let T:V → W be a linear transformation between vector spaces. Then ker(T)=T-1(0).TrueFalse
Please answer the following question. Thank you. 30. Let T:V W be a linear transformation from a vector space V into a vector space W.Prove that the range of T is a subspace of W.[ Hint: Typical elements of the range have the form T(x) and T(w) for some x, w in V.]
Find an example of a vector space V, and a linear transformation T : V + V such that R(T) = ker(T). Your vector space V must have dimension > 2. You may find it helpful to let V be a euclidean space and T a matrix transformation,
Exercise 7.2.9 Let T:V → V be a linear transformation where V is finite dimensional. Show that exactly one of (i) and (ii) holds: (i) T(v) = 0 for some v + 0 in V; (ii) T(x) = v has a solution x in V for every v in V.
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 T is an isomorphism, show that T-1 is the unique generalized inverse of T.
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...
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 SoToS = S. If V and W are finite dimensional, show that there exists a generalized inverse of T.
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.
4.10. Let T be a linear transformation on a vector space V satisfying T-T2 = id. Show that T is invertible.