**Vector Spaces****Prove property 3 by rearranging the following:****Property 3:** For every element \( A, B, C \) in \( V = M_{3 \times 3} \), \((A \oplus B) \oplus C = A \oplus (B \oplus C)\).---**A)** Let \( a_{ij}, b_{ij}, c_{ij}, d_{ij}, e_{ij} \) be the \( i,j \)-th entries in \( A, B, C, D = (A \oplus B) \oplus C, E = A \oplus (B \oplus C) \), respectively.**B)** Suppose \( A, B, \) and \( C \) are in \( V = M_{3 \times 3} \).**C)** From the definition of \( A \oplus B \), \( d_{ij} = (a_{ij} \times b_{ij}) \times c_{ij} \) and \( e_{ij} = a_{ij} \times (b_{ij} \times c_{ij}) \).**D)** So, \((A \oplus B) \oplus C = A \oplus (B \oplus C) \).**E)** Since \( a_{ij}, b_{ij}, c_{ij} \) are real numbers, \( d_{ij} = e_{ij} \).*Answer for example FE, D, C, B, A*[Box] [Box] [Box] [Box] [Box]---This document proves an associative property of vector spaces for matrices \( M_{3 \times 3} \) using element-wise operations and the associativity of real number multiplication.

Given:Property 3: For every element A, B, C in V=M3×3, A⊕B⊕C=A⊕B⊕C. We have to prove property 3 by…

Vector Spaces Prove property 3 by rearranging the flig Property 3 For every element A, B, C in V=M3x3 (ABB) C= A (BDC) A tet aij, bij, cij, dij, ei; and the with entries in A₁B₁C₁D = (AⓇB) + C, E = A (BOC), respectively B) Suppose A, B and C are in V = M3x3 c) from the definition of AB, dij -={a; jxb;;) x c;; and ei; = aijx (bijx cij) D) So, (AB) C = A + (PⓇC Since aij, bij, ci are real #'s dij=eij

Vector Spaces Prove property 3 by rearranging the flig Property 3 For every element A, B, C in V=M3x3 (ABB) C= A (BDC) A tet aij, bij, cij, dij, ei; and the with entries in A₁B₁C₁D = (AⓇB) + C, E = A (BOC), respectively B) Suppose A, B and C are in V = M3x3 c) from the definition of AB, dij -={a; jxb;;) x c;; and ei; = aijx (bijx cij) D) So, (AB) C = A + (PⓇC Since aij, bij, ci are real #'s dij=eij

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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