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
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Vector Spaces**
**Prove property 3 by rearranging the following:**
**Property 3:** For every element \( A, B, C \) in \( V = M_{3 \times 3} \), \((A \oplus B) \oplus C = A \oplus (B \oplus C)\).
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**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]
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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe0243d8a-eca1-425a-958c-12b2dd6f68ba%2F49dc4b3e-6d0b-4590-9a6a-51c001d28f55%2F76u8k8i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Vector Spaces**
**Prove property 3 by rearranging the following:**
**Property 3:** For every element \( A, B, C \) in \( V = M_{3 \times 3} \), \((A \oplus B) \oplus C = A \oplus (B \oplus C)\).
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**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]
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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.
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