Linear algebra 5. (a) Let V be an inner product space with F = R, and let T : V → V be an isomorphism of innerproduct spaces. Show that if ceR is an eigenvalue of T, then c= 1 or c = -1.(b) Suppose V is a finite-dimensional inner product space with Fa linear transformation such that V has an orthonormal basis {v1, v2, . .. , Vn} of eigenvectors ofT. Furthermore, assume that all eigenvalues of T come from the set {-1,1}. Prove that T is anisomorphism of inner product spaces.R, and suppose T : V → V is

For the first question we will use the key property of isomorphism that the linear transform if the…

Answered: (a) Let V be an inner product space…

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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Linear algebra

5. (a) Let V be an inner product space with F = R, and let T : V → V be an isomorphism of inner
product spaces. Show that if ceR is an eigenvalue of T, then c= 1 or c = -1.
(b) Suppose V is a finite-dimensional inner product space with F
a linear transformation such that V has an orthonormal basis {v1, v2, . .. , Vn} of eigenvectors of
T. Furthermore, assume that all eigenvalues of T come from the set {-1,1}. Prove that T is an
isomorphism of inner product spaces.
R, and suppose T : V → V is

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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