How can this vector be determined?

I am currently preparing for the 2024 Math Abitur exam.

I have absolutely no idea how to solve problem 3.1.2. Could someone please explain the solution to me?

The attachment contains both the task and the solution, but despite the solution document I don't understand it.

Task:

Solution:

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Halbrecht
1 year ago

the approach should be clear

two equations which arise from the first line of the matrix and the first of the vector.

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Both equations are brought to form “=0 “

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Normally: Two Glg allows finding the unknowns x1 and x2

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Here, however, the solution method leads to 0=0

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Therefore you can choose x1 or x2 free

.

.

PS

changing an entry in the matrix you would have found unambiguous x1 and x2

Halbrecht
1 year ago
Reply to  Julian255

a vector !

In addition, S has the form 2 times2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .