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\title{
Exam for ``Linear Algebra 4'' (6LAG4) \\ summer semester 2025/2026
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\section*{Questions set no. 4}

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\textbf{1.}
Describe the Fano plane: what kind of structure it is, its properties,
why it is remarkable.

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\textbf{2.}
The group of motions of an Euclidean affine space: definition, properties, 
relation to the orthogonal group.

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\textbf{3.}

\textbf{1)}
Let $a_1, \dots, a_n$ be points in an affine space. Does the expression 
$a_1 - a_2 + a_3 - \dots + (-1)^{n+1} a_n$ (alternating signs sum) make sense?

\textbf{2)}
Let $a_1, a_2, a_3$ be points in an affine space. Describe geometrically the
set of all barycentric combinations of these points with positive coefficients.



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