Existence of a qudratic form
Task number: 4447
Decide, whether in the vector space \(\mathbb R^3\) there exists a quadratic form \(g\) for which the following holds:
\(g((1, 0, 0)^T) = 1\), \( g((0, 1, 0)^T) = 2\), \( g((0, 0, 1)^T) = 3\), \(g((1, 0, 1)^T) = 4\), \( g((1, 1, 0)^T) = 5\), \(g((0, 1, 1)^T) = 6\) and \(g((1, 1, 1)^T) = 7\).
Hint
Try first to construct the matrix of the symmetric bilinear form associated with \(g\).
How much data is needed to construct the matrix?
Solution
We use the relation from the definition of the matrix of the form: \( b_{ij}=\frac12 (g(v_i+v_j)-g(v_i)-g(v_j)) \).
From the first six entries, we construct the only possible matrix of the form \(g\) with respect to the standard basis \( \boldsymbol B = \begin{pmatrix} 1 & 1 & 0 \\ 1 & 2 & 1/2 \\ 0 & 1/2 & 3 \end{pmatrix} \).
However, the form corresponding to this matrix does not satisfy the last condition that \(g((1{,}1,1)^T)=7\), because the product with the matrix yields 9 instead of 7.
Answer
There is no quadratic form that satisfies these seven conditions.
