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Let V be a finite dimensional vector space, T = L(V), and T² = T. (a) Prove that im(T) ^ ker(T) = {0}. (b) Prove that V = im(T) + ker(T). (c) Let V = F. Prove that there is a basis of V such that the matrix of T with respect to this basis is a diagonal matrix whose entries are all O's or 1's.

Let V be a finite dimensional vector space, T = L(V), and T² = T. (a) Prove that im(T) ^ ker(T) = {0}. (b) Prove that V = im(T) + ker(T). (c) Let V = F. Prove that there is a basis of V such that the matrix of T with respect to this basis is a diagonal matrix whose entries are all O's or 1's.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.3: Change Of Basis
Problem 22EQ
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Let V be a finite dimensional vector space, T = L(V), and T² = T. (a) Prove that im(T) ker(T) = {0}. (b) Prove that V = im(T) + ker(T). (c) Let V = Fr. Prove that there is a basis of V such that the matrix of T with respect to this basis is a diagonal matrix whose entries are all 0's or 1's.
Transcribed Image Text:Let V be a finite dimensional vector space, T = L(V), and T² = T. (a) Prove that im(T) ker(T) = {0}. (b) Prove that V = im(T) + ker(T). (c) Let V = Fr. Prove that there is a basis of V such that the matrix of T with respect to this basis is a diagonal matrix whose entries are all 0's or 1's.
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