4.5.17 Let V denote R^2 equipped with the l^ 1 norm and let W denote R^2 equipped with the l^∞ norm. Show that the linear map in L(V,W) represented by [ 1 1 ] is an isometry. Remark: On the other hand, if n ≥ 3, then R^n with the l^1 norm and R^n with the l^∞ norm are not isometric.
[ 1 -1 ]
4.5.17 Let V denote R^2 equipped with the l^ 1 norm and let W denote R^2...
4. Let L: V→ W be a linear map. Let w be an element of W. Let uo be an ele- ment of V such that LvO-w. Show that any solution of the equation L(X)-w is of type uo + u, where u is an element of the kernel of L.
Let V and W be a vector spaces over F and T ∈ L(V, W) be invertible. Prove that T-1 is also linear map from W to V . Please show all steps, thank you
2. For each p> 1, denote by || . || the norm in R" defined by: n || 2 || p = [f(x)}; where f(x) = 5:), Vx = ((1, ... , In) € R". i=1 For p > 1 and r* in the dual of R", consider the following optimization problem: n(x*) = sup|(2*, 2) : ||- ||| < 1] = sup[(x*, r) : f(x) < 1] 1 . Prove that n(x*) = || 2* ||g, where q> 1...
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 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.
Let and let T : R4 → R4 be the map T(v) = v-2 proju,V-2 proju, w (a) Show that T is a linear transformation. (b) Find T] and (c) Show that T is invertible and find T-1 Let and let T : R4 → R4 be the map T(v) = v-2 proju,V-2 proju, w (a) Show that T is a linear transformation. (b) Find T] and (c) Show that T is invertible and find T-1
Problem 13.5. Let V and W be inner product spaces and T є L(V : W). Let(..) v and (..)w denote their respective mner products. Let ui, , uk be an orthonormal basts o V and W1,…,wn an orthonormal bass o W. Let A and A* be the matrices representing T and T with respect to the given bases. Show that A. = A i.e., A. is obtained from A by taking the transpose and conjugating all the entries (in...
3. Let TEL(V,W), and assume that S E L(W) is an isometry. Prove that T and ST have the same singular values.
solution to 2 (ii) Show that the image of f is not a subspace of R 2. Let U, V, and W be vector spaces over the field k, and let f: Ux V- W be a bilinear map. Show that the image of f is a union of subspaces of W. 3. Let k be a field, and let U, V, and W be vector spaces over k. Recall that (ii) Show that the image of f is not...