Here's a difficult problem in an area I'm not at all familiar with.(adsbygoogle = window.adsbygoogle || []).push({});

A product * is defined on R[tex]^{4}[/tex] in the following way:

(a, b, c, d)*(a', b', c', d') = (cd' - c'd, ac' - a'c + cb' - c'b, a'd - ad' + bd' - b'd, c'd - cd')

Find all subsets S of R[tex]^{4}[/tex] that satisfies the following two conditions:

1: For all x,y [tex]\in[/tex] S and all a,b [tex]\in[/tex] R, ax + by [tex]\in[/tex] S

2: For all x [tex]\in[/tex] S and all y [tex]\in[/tex] R[tex]^{4}[/tex], x * y [tex]\in[/tex] S

Condtion 1 seems to me to say we're on a 2D-plane in 4-space but the strange symmetry of the product must restrict this in some way. Condition 2 seems to be about linearity within S somehow. I'm really new at this and would really appreciate quite a lot of help with it.

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# Difficult Problem

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