Let K = R be the field of real numbers. Define the algebra of quaternions H. Prove that H is isomorphic to the algebra A with basis {1, x, y, z} and multi- plication given by letting 1 be the unit and 8 X 2x-2 y x+y-z Z x+y+z-2 Y x+y+z-2 2y-2 −x + y + z Z x - y + z x+y+z-2 2z-2

Let K = R be the field of real numbers. Define the algebra of quaternions H. Prove that H is isomorphic to the algebra A with basis {1, x, y, z} and multi- plication given by letting 1 be the unit and 8 X 2x-2 y x+y-z Z x+y+z-2 Y x+y+z-2 2y-2 −x + y + z Z x - y + z x+y+z-2 2z-2

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.5: Basis And Dimension

Problem 69E: Find a basis for R2 that includes the vector (2,2).

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Question

Let K = R be the field of real numbers. Define the algebra of quaternions H.
Prove that H is isomorphic to the algebra A with basis {1, x, y, z} and multi-
plication given by letting 1 be the unit and
8
X
2x-2
y x+y-z
Z
x+y+z-2
Y
x+y+z-2
2y-2
−x + y + z
Z
x - y + z
x+y+z-2
2z-2

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