Let M be the space of m x n matrices, Z the subspace of all matrices m e M such that the sum of all the entries of m is zero. Let R denote the subspace of M consisting of all matrices m such that the sum of the entries in every row of m is zero. Let C denote the subspace of M consisting of all matrices m such that the sum of the entries in every column of m is zero. Prove that dim(Z) = mn – 1, dim(R) = mn – m and dim(C) = mn – n. nn- Prove that Z = R+C and from this conclude that dim(RnC) = (m–1)(n – 1). %3D
Let M be the space of m x n matrices, Z the subspace of all matrices m e M such that the sum of all the entries of m is zero. Let R denote the subspace of M consisting of all matrices m such that the sum of the entries in every row of m is zero. Let C denote the subspace of M consisting of all matrices m such that the sum of the entries in every column of m is zero. Prove that dim(Z) = mn – 1, dim(R) = mn – m and dim(C) = mn – n. nn- Prove that Z = R+C and from this conclude that dim(RnC) = (m–1)(n – 1). %3D
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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