Let V be a finite-dimensional vector space and T : V → V be a linear transformation. (a) Suppose V = R(T) + N(T). Show that V = R(T) ⊕ N(T). (b) Suppose that R(T) ∩ N(T) = {0}. Show that V = R(T) ⊕ N(T).

Let V be a finite-dimensional vector space and T : V → V be a linear transformation. (a) Suppose V = R(T) + N(T). Show that V = R(T) ⊕ N(T). (b) Suppose that R(T) ∩ N(T) = {0}. Show that V = R(T) ⊕ N(T).

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Linear Transformations

Section6.1: Introduction To Linear Transformations

Problem 33E: Linear Transformation Given by a Matrix In Exercises 33-38, define the linear transformations...

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Question

Linear Algebra - Linear Transformation

Let V be a finite-dimensional vector space and T : V → V be a linear transformation.

(a) Suppose V = R(T) + N(T). Show that V = R(T) ⊕ N(T).

(b) Suppose that R(T) ∩ N(T) = {0}. Show that V = R(T) ⊕ N(T).

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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