Elements Of Modern Algebra

8th Edition

Gilbert + 2 others

ISBN: 9781285463230

Textbook Problem

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a. Prove part a. of Theorem 1.34.

b. Prove part b. of Theorem 1.34.

Theorem 1.34 □ Distributive Properties

Let A be an m × n matrix over R , let B and C be n × p matrices over R , and let D be a p × q matrix over R . Then

A ( B + C ) = A B + A C , and

( B + C ) D = B D + C D

(a)

To determine

To prove: Let A be m×n matrix in ℝ, and let B and C are n×p matrices over ℝ. Then, A(B+C)=AB+AC.

Explanation

Given information:

A be m×n matrix in ℝ, and B, C are n×p matrices over ℝ.

Formula used:

(1) Addition in Mm × n(ℝ) is defined by

[aij]m × n+[bij]m × n=[cij]m × n, where cij=aij+bij.

(2) Definition of matrix multiplication:

The product of m×n matrix A over ℝ and n×p matrix B over ℝ is m×p matrix C=AB, where the element cij in row i and column j of AB is found by using the elements in row i of A, and the elements in column j of B in the following manner:

column j of B column j of Crow i of A[⋮⋮⋮⋮ai1ai2ai3⋯ain⋮⋮⋮⋮]⋅[⋯b1j⋯⋯b2j⋯⋯b3j⋯⋮⋯bnj⋯]=[⋮⋯cij⋯⋮]row i of C

Where cij=ai1b1j+ai2b2j+ ⋯ + ainbnj.

Proof:

Let A=[aik]m × n, B=[bkj]n × p and C=[ckj]n × p are matrices over ℝ

(b)

To determine

To prove: Let B and C are n×p matrices over ℝ, and let D be a p×q matrix over ℝ. Then (B+C)D=BD+CD.

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Sect-1.1 P-1TFE Sect-1.1 P-2TFE Sect-1.1 P-3TFE Sect-1.1 P-4TFE Sect-1.1 P-5TFE Sect-1.1 P-6TFE Sect-1.1 P-7TFE Sect-1.1 P-8TFE Sect-1.1 P-9TFE Sect-1.1 P-10TFE

Sect-1.1 P-1E Sect-1.1 P-2E Sect-1.1 P-3E Sect-1.1 P-4E Sect-1.1 P-5E Sect-1.1 P-6E Sect-1.1 P-7E Sect-1.1 P-8E Sect-1.1 P-9E Sect-1.1 P-10E Sect-1.1 P-11E Sect-1.1 P-12E Sect-1.1 P-13E Sect-1.1 P-14E Sect-1.1 P-15E Sect-1.1 P-16E Sect-1.1 P-17E Sect-1.1 P-18E Sect-1.1 P-19E Sect-1.1 P-20E Sect-1.1 P-21E Sect-1.1 P-22E Sect-1.1 P-23E Sect-1.1 P-24E Sect-1.1 P-25E Sect-1.1 P-26E Sect-1.1 P-27E Sect-1.1 P-28E Sect-1.1 P-29E Sect-1.1 P-30E Sect-1.1 P-31E Sect-1.1 P-32E Sect-1.1 P-33E Sect-1.1 P-34E Sect-1.1 P-35E Sect-1.1 P-36E Sect-1.1 P-37E Sect-1.1 P-38E Sect-1.1 P-39E Sect-1.1 P-40E Sect-1.1 P-41E Sect-1.1 P-42E Sect-1.1 P-43E Sect-1.2 P-1TFE Sect-1.2 P-2TFE Sect-1.2 P-3TFE Sect-1.2 P-4TFE Sect-1.2 P-5TFE Sect-1.2 P-6TFE Sect-1.2 P-7TFE Sect-1.2 P-8TFE Sect-1.2 P-9TFE Sect-1.2 P-1E Sect-1.2 P-2E Sect-1.2 P-3E Sect-1.2 P-4E Sect-1.2 P-5E Sect-1.2 P-6E Sect-1.2 P-7E Sect-1.2 P-8E Sect-1.2 P-9E Sect-1.2 P-10E Sect-1.2 P-11E Sect-1.2 P-12E Sect-1.2 P-13E Sect-1.2 P-14E Sect-1.2 P-15E Sect-1.2 P-16E Sect-1.2 P-17E Sect-1.2 P-18E Sect-1.2 P-19E Sect-1.2 P-20E Sect-1.2 P-21E Sect-1.2 P-22E Sect-1.2 P-23E Sect-1.2 P-24E Sect-1.2 P-25E Sect-1.2 P-26E Sect-1.2 P-27E Sect-1.2 P-28E Sect-1.3 P-1TFE Sect-1.3 P-2TFE Sect-1.3 P-3TFE Sect-1.3 P-4TFE Sect-1.3 P-5TFE Sect-1.3 P-6TFE Sect-1.3 P-1E Sect-1.3 P-2E Sect-1.3 P-3E Sect-1.3 P-4E Sect-1.3 P-5E Sect-1.3 P-6E Sect-1.3 P-7E Sect-1.3 P-8E Sect-1.3 P-9E Sect-1.3 P-10E Sect-1.3 P-11E Sect-1.3 P-12E Sect-1.4 P-1TFE Sect-1.4 P-2TFE Sect-1.4 P-3TFE Sect-1.4 P-4TFE Sect-1.4 P-5TFE Sect-1.4 P-6TFE Sect-1.4 P-7TFE Sect-1.4 P-8TFE Sect-1.4 P-9TFE Sect-1.4 P-1E Sect-1.4 P-2E Sect-1.4 P-3E Sect-1.4 P-4E Sect-1.4 P-5E Sect-1.4 P-6E Sect-1.4 P-7E Sect-1.4 P-8E Sect-1.4 P-9E Sect-1.4 P-10E Sect-1.4 P-11E Sect-1.4 P-12E Sect-1.4 P-13E Sect-1.4 P-14E Sect-1.4 P-15E Sect-1.4 P-16E Sect-1.5 P-1TFE Sect-1.5 P-2TFE Sect-1.5 P-3TFE Sect-1.5 P-1E Sect-1.5 P-2E Sect-1.5 P-3E Sect-1.5 P-4E Sect-1.5 P-5E Sect-1.5 P-6E Sect-1.5 P-7E Sect-1.5 P-8E Sect-1.5 P-9E Sect-1.5 P-10E Sect-1.6 P-1TFE Sect-1.6 P-2TFE Sect-1.6 P-3TFE Sect-1.6 P-4TFE Sect-1.6 P-5TFE Sect-1.6 P-6TFE Sect-1.6 P-7TFE Sect-1.6 P-8TFE Sect-1.6 P-9TFE Sect-1.6 P-10TFE Sect-1.6 P-11TFE Sect-1.6 P-12TFE Sect-1.6 P-1E Sect-1.6 P-2E Sect-1.6 P-3E Sect-1.6 P-4E Sect-1.6 P-5E Sect-1.6 P-6E Sect-1.6 P-7E Sect-1.6 P-8E Sect-1.6 P-9E Sect-1.6 P-10E Sect-1.6 P-11E Sect-1.6 P-12E Sect-1.6 P-13E Sect-1.6 P-14E Sect-1.6 P-15E Sect-1.6 P-16E Sect-1.6 P-17E Sect-1.6 P-18E Sect-1.6 P-19E Sect-1.6 P-20E Sect-1.6 P-21E Sect-1.6 P-22E Sect-1.6 P-23E Sect-1.6 P-24E Sect-1.6 P-25E Sect-1.6 P-26E Sect-1.6 P-27E Sect-1.6 P-28E Sect-1.6 P-29E Sect-1.6 P-30E Sect-1.6 P-31E Sect-1.6 P-32E Sect-1.7 P-1TFE Sect-1.7 P-2TFE Sect-1.7 P-3TFE Sect-1.7 P-4TFE Sect-1.7 P-5TFE Sect-1.7 P-6TFE Sect-1.7 P-1E Sect-1.7 P-2E Sect-1.7 P-3E Sect-1.7 P-4E Sect-1.7 P-5E Sect-1.7 P-6E Sect-1.7 P-7E Sect-1.7 P-8E Sect-1.7 P-9E Sect-1.7 P-10E Sect-1.7 P-11E Sect-1.7 P-12E Sect-1.7 P-13E Sect-1.7 P-14E Sect-1.7 P-15E Sect-1.7 P-16E Sect-1.7 P-17E Sect-1.7 P-18E Sect-1.7 P-19E Sect-1.7 P-20E Sect-1.7 P-21E Sect-1.7 P-22E Sect-1.7 P-23E Sect-1.7 P-24E Sect-1.7 P-25E Sect-1.7 P-26E Sect-1.7 P-27E Sect-1.7 P-28E Sect-1.7 P-29E Sect-1.7 P-30E

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MathAlgebraElements Of Modern Algebra8th Edition

Elements Of Modern Algebra

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Elements Of Modern Algebra

a. Prove part a. of Theorem 1.34. b. Prove part b. of Theorem 1.34. Theorem 1.34 □ Distributive Properties Let A be an m × n matrix over R , let B and C be n × p matrices over R , and let D be a p × q matrix over R . Then A ( B + C ) = A B + A C , and ( B + C ) D = B D + C D

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Math
Algebra Elements Of Modern Algebra 8th Edition

a. Prove part a. of Theorem 1.34.

b. Prove part b. of Theorem 1.34.

Theorem 1.34 □ Distributive Properties

Let A be an m × n matrix over R , let B and C be n × p matrices over R , and let D be a p × q matrix over R . Then

A ( B + C ) = A B + A C , and

( B + C ) D = B D + C D