Section VIII.1

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Problem VIII.1.1

In some Cartesian coordinate system of a four-dimensional affine space a plane is given by the system of equations

ξ1+ξ2+ξ3+ξ4=12ξ1+3ξ2+4ξ3+5ξ4=−1 \begin{aligned} \xi ^{1}+\xi ^{2}+\xi ^{3}+\xi ^{4} & =1 \\ 2 \xi ^{1}+3 \xi ^{2}+4 \xi ^{3}+5 \xi ^{4} & =-1 \end{aligned}

Write its parametric equations (find the initial point and a basis in the direction subspace).

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Problem VIII.1.2
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(a)

What can the intersection of two planes be?

(b)

In an nn-dimensional affine space, estimate the dimension of the plane obtained as the intersection of planes of dimensions k1k_{1} and k2k_{2}.

Problem VIII.1.3

Prove that in an affine space any two lines lie in some three-dimensional plane.

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