Let Z be the subspace of all m x n matrices such that the sum of all its entries is zero. Let R be the subspace of all m x n matrices such that the sum of each row is zero. Let C behe subspace of all m x n matrices such that the sum of each column is zero. Determine, with a proof, dim(Z), dim(R), dim(C), dim(R+C), dim(RnC).

Let Z be the subspace of all m x n matrices such that the sum of all its entries is zero. Let R be the subspace of all m x n matrices such that the sum of each row is zero. Let C behe subspace of all m x n matrices such that the sum of each column is zero. Determine, with a proof, dim(Z), dim(R), dim(C), dim(R+C), dim(RnC).

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

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

Let Z be the subspace of all m x n matrices such that the sum of all its entries
is zero.
Let R be the subspace of all m x n matrices such that the sum of each row is
zero.
Let C behe subspace of all m x n matrices such that the sum of each column is
zero.
Determine, with a proof, dim(Z), dim(R), dim(C), dim(R +C), dim(RnC).

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