Pespectives in Mathematical Sciences Due: Tuesday, June 9, 2020, on NUCT.
Problem 1. LetHbe the division ring of quaternions defined Example 1.3 (4) of Lecture 1, and letV = (R4,+,·) be the leftH-vector space with “+” given by the usual vectorsum inR4 and with scalar multiplication ·:H×R4→R4 defined by
(a+ib+jc+kd)·
x1
x2
x3 x3
=
a −b −c −d
b a −d c
c d a −b
d −c b a
x1
x2
x3 x3
.
Prove the following statements:
(a) The family (v) consisting of the single vector
v=
1 0 0 0
is a linearly independent family in the leftH-vector spaceV. (b) The family (v) generates the left H-vector spaceV.
(c) The family (v) is a basis leftH-vector spaceV.
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