Definition:Let L : V –→ V be a linear transformation of a vector space V. A subspace U of V is said to beL-invariant if L(U) C U.6. (*) Let L :V → V be a linear transformation and suppose that V has dimension n.a) If U1,Us are L-invariant subspaces, show that their sumU1 + U2 + · ·· +Ug = {u1 + U2 + · ··+ us : U; E U¡}..is L-invariant.b) Suppose that L is invertible and U is an L-invariant subspace. For all v 4 U, show that L(v) ¢ U.c) If U is a L-invariant subspace of dimension m, prove that there is a basis B for V so that thematrix of L with respect to B has the following shape:AВ[L]g = |where A, B and C are matrices, and 0 denotes an all zero matrix. Specify the number of rowsand columns in each of A, B, C and the all zero matrix you find.

6. a. Given that each Ui's are L invariant sub spaces of V, then we define a map as follows. Let…

Definition: Let L : V →→ V be a linear transformation of a vector space V. A subspace U of V is said to be L-invariant if L(U) C U. 6. (*) Let L: V → V be a linear transformation and suppose that V has dimension n. a) If U1, Us are L-invariant subspaces, show that their sum ... U1 + U2 + · · · + Us = {u1 +2 + · · · + us : U; E U¡} %3D is L-invariant. b) Suppose that L is invertible and U is an L-invariant subspace. For all v ¢ U, show that L(v) ¢ U. c) If U is a L-invariant subspace of dimension m, prove that there is a basis B for V so that the matrix of L with respect to B has the following shape: 1 A В [L]s = | C where A, B and C are matrices, and 0 denotes an all zero matrix. Specify the number of rows and columns in each of A, B, C and the all zero matrix you find.

Definition: Let L : V →→ V be a linear transformation of a vector space V. A subspace U of V is said to be L-invariant if L(U) C U. 6. (*) Let L: V → V be a linear transformation and suppose that V has dimension n. a) If U1, Us are L-invariant subspaces, show that their sum ... U1 + U2 + · · · + Us = {u1 +2 + · · · + us : U; E U¡} %3D is L-invariant. b) Suppose that L is invertible and U is an L-invariant subspace. For all v ¢ U, show that L(v) ¢ U. c) If U is a L-invariant subspace of dimension m, prove that there is a basis B for V so that the matrix of L with respect to B has the following shape: 1 A В [L]s = | C where A, B and C are matrices, and 0 denotes an all zero matrix. Specify the number of rows and columns in each of A, B, C and the all zero matrix you find.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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