Quadratic Equations

To solve the problem, we analyze the set and the properties of quadratic forms and linear systems associated with elements .

Given:
So is the set of coefficients defining a positive definite quadratic form.

Recall:
For the quadratic form the form is positive definite if and only if: and (since it must be positive for all ).

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1. Check if :
Calculate the determinant: Since the determinant is negative, the quadratic form is not positive definite.
Statement 1 is FALSE.

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2. If , then :
Conditions for positive definiteness:
and Multiply both sides by 2: Statement 2 is TRUE.

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3. For any , the system:
has a unique solution.

Analysis:
The coefficient matrix is: Since , is positive definite, so .
Therefore, is invertible and the system has a unique solution.
Statement 3 is TRUE.

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4. For any , the system:
has a unique solution.

Analysis:
Coefficient matrix: Determinant: Since , clearly .
Also, from positive definiteness of , .
We want to check if . Let's analyze: Since and , sum is positive.
Therefore, and is invertible.
Hence, the system has a unique solution (only trivial if right-hand side is zero).
Statement 4 is TRUE.

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Summary of Truth Values:
- Statement 1: FALSE
- Statement 2: TRUE
- Statement 3: TRUE
- Statement 4: TRUE

Final Answer: