Chapter VIII

Chapter VIII

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Section VIII.1§ 1[3/100%]
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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Section VIII.2§ 2[3/100%]
Problem VIII.2.1

Reduce to canonical form the equation

2(ξ2)2−3(ξ3)2−23ξ1ξ2−4ξ1ξ3+43ξ2ξ3+50ξ3=80. 2\left(\xi ^{2}\right)^{2}-3\left(\xi ^{3}\right)^{2}-2 \sqrt{3} \xi ^{1} \xi ^{2}-4 \xi ^{1} \xi ^{3}+4 \sqrt{3} \xi ^{2} \xi ^{3}+50 \xi ^{3}=80 .
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Problem VIII.2.2

Without reducing the equation to canonical form, determine which second-order surface it defines:

(ξ1)2+4ξ1ξ2+6ξ1ξ3−(ξ2)2+2ξ2ξ3+4(ξ3)2+2ξ1=0. \left(\xi ^{1}\right)^{2}+4 \xi ^{1} \xi ^{2}+6 \xi ^{1} \xi ^{3}-\left(\xi ^{2}\right)^{2}+2 \xi ^{2} \xi ^{3}+4\left(\xi ^{3}\right)^{2}+2 \xi ^{1}=0 .
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Problem VIII.2.3

For which values of the parameter aa is the surface with the equation

(ξ1)2+(ξ2)2+(ξ3)2+2a(ξ1ξ2+ξ2ξ3+ξ1ξ3)+4a=0 \left(\xi ^{1}\right)^{2}+\left(\xi ^{2}\right)^{2}+\left(\xi ^{3}\right)^{2}+2 a\left(\xi ^{1} \xi ^{2}+\xi ^{2} \xi ^{3}+\xi ^{1} \xi ^{3}\right)+4 a=0

an ellipsoid?

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