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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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.
Prove that Z = R+C and from this conclude that dim(RNC) = (m– 1)(n–1).

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