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Pespectives in Mathematical Sciences Due: Tuesday, June 9, 2020, on NUCT. Problem 1. Let H be the division ring of quaternions defined Example 1.3 (4) of Lecture 1, and let V = (R

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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.

1

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