Page 457 - Advanced Linear Algebra

Affine Geometry 441

13. Let dim²= ³ ~ . Prove the following:
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a The join of any two distinct points is a line.
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b The intersection of any two nonparallel lines is a point.
14. Let dim²= ³ ~ . Prove the following:
)
a The join of any two distinct points is a line.
)
b The intersection of any two nonparallel planes is a line.
c The join of any two lines whose intersection is a point is a plane.
)
d The intersection of two coplanar nonparallel lines is a point.
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e The join of any two distinct parallel lines is a plane.
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f The join of a line and a point not on that line is a plane.
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g The intersection of a plane and a line not on that plane is a point.
)
15. Prove that ¢ = ¦ = is a surjective affine transformation if and only if
B
~ k ; $ for some $  = and  ²= ³.
16. Verify the group-theoretic remarks about the group homomorphism
B¢ aff
²=³ ¦ ²=³ and the subgroup trans ²=³ of aff ²=³.