University:
University of Michigan
Course:
MATH 116 | Calculus II
Academic year:

2024
Views:

490

Pages:

1
Author:

banenetta0l

Question 1 What is meant by the term commutative in algebra? Explain along with an example. Answer Definition: The commutative property in algebra states that order does not matter. In other words, if ∗ ∗ is an operation and 𝑎 ∗ 𝑏 = 𝑏 ∗ 𝑎 a∗b=b∗a for all 𝑎 a and 𝑏 b, then ∗ ∗ is commutative. Example (Commutative): Addition: 𝑎 + 𝑏 = 𝑏 + 𝑎 a+b=b+a. Multiplication: 𝑎 𝑏 = 𝑏 𝑎 ab=ba. Example (Non-Commutative): Subtraction: 𝑎 − 𝑏 ≠ 𝑏 − 𝑎 a−b  =b−a. Division: ( 𝑎 𝑏 ) ≠ ( 𝑏 𝑎 ) . ( b a )  =( a b ). Question 2 Show that quaternion multiplication is not commutative. That is, give an example to show that sometimes 𝑞 1 𝑞 2 ≠ 𝑞 2 𝑞 1 q 1 q 2  =q 2 q 1 . Answer Quaternions form a finite-dimensional vector space over 𝑅 R. Suppose that their multiplication is commutative. Then, it becomes a field. It implies that they are a finite-dimensional field extension of the real numbers. But according to the fundamental theorem of algebra, there exists only one such extension, which is the complex numbers. That is, if quaternion multiplication was commutative, then it would just reduce to the complex numbers. Quaternion multiplication is used to model three-dimensional rotations. According to the multiplication, 𝑖 ⋅ 𝑗 = 𝑘 , 𝑗 ⋅ 𝑖 = − 𝑘 . i⋅j=k,j⋅i=−k. That is, rotations in the three dimensions are not commutative. Example: Consider two quaternions 𝑞 1 = 1 2 𝑖 + 1 2 𝑗 , 𝑞 2 = 1 2 𝑗 + 1 2 𝑘 . q 1 = 2 1 i+ 2 1 j,q 2 = 2 1 j+ 2 1 k. On multiplying 𝑞 1 q 1 and 𝑞 2 q 2 in both ways: 𝑞 1 𝑞 2 = ( 1 2 𝑖 + 1 2 𝑗 ) ( 1 2 𝑗 + 1 2 𝑘 ) = 1 2 𝑖 𝑗 + 1 2 𝑖 𝑘 + 1 2 𝑗 𝑗 + 1 2 𝑗 𝑘 = 1 2 𝑘 + 1 2 𝑖 + 1 2 ( − 1 ) + 1 2 𝑗 . q 1 q 2 =( 2 1 i+ 2 1 j)( 2 1 j+ 2 1 k)= 2 1 ij+ 2 1 ik+ 2 1 jj+ 2 1 jk= 2 1 k+ 2 1 i+ 2 1 (−1)+ 2 1 j. Reversing the multiplication: 𝑞 2 𝑞 1 = ( 1 2 𝑗 + 1 2 𝑘 ) ( 1 2 𝑖 + 1 2 𝑗 ) = 1 2 𝑗 𝑖 + 1 2 𝑗 𝑗 + 1 2 𝑘 𝑖 + 1 2 𝑘 𝑗 = 1 2 ( − 𝑘 ) + 1 2 ( − 1 ) + 1 2 𝑗 + 1 2 𝑖 . q 2 q 1 =( 2 1 j+ 2 1 k)( 2 1 i+ 2 1 j)= 2 1 ji+ 2 1 jj+ 2 1 ki+ 2 1 kj= 2 1 (−k)+ 2 1 (−1)+ 2 1 j+ 2 1 i. Thus, quaternion multiplication is not commutative.