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Linear algebra and geometry

Linear algebra and geometry

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Let be an endomorphism with characteristic polynomial .

Which one of the following statements is true?

If the linear system with 4 equations and 4 variables has solutions with , then necessarily

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If the linear system with 5 equations and 5 variables has solutions with , then necessarily

Let us consider the linear system with 4 variables depending on the real parameter :

Find the right statement.

If the system admits only one solutions.

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If the system admits solutions.

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If the system admits solutions.

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For any the system does not admit any solution.

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If the linear system with 4 equations and 5 variables has solutions with , then necessarily

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Let the versors of the coordinate axes of . For every , let us consider the vectors

Which of the following statements is true?

There exist exactly two values of such that the vectors are linearly independent.

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There are exactly two values of such that the vectors are coplanar.

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For every the vectors are coplanar.

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For every the vectors are linearly independent.

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Consider the points and and the lines

For all points , the triangle of vertices , and has area equal to .

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There does not exist a line such that and .

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For all points , , and are collinear.

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There exists , such that , and are collinear

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Let the versors of the coordinate axes of . For every , let us consider the vectors

Which of the following statements is true?

For every the vectors are coplanar.

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There exists a unique such that the vectors are coplanar.

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For every the vectors are linearly independent.

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There exists a unique such that the vectors are linearly independent.

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In the euclidean space let us consider the points

Which of the following statements is true?

There exists exactly two such that the points are coplanar.

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For every the points are collinear.

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For every the points are coplanar.

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There exists exactly two such that the points are collinear.

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Consider the points and and the lines

For all points , the triangle of vertices , and has area equal to .

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There does not exist a line such that and .

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If , then , and are collinear.

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There exists a unique point such that , and are collinear.

❌

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