Analytic expression for basis change

Task number: 2740

The quadratic form \(g\) on the vector space \(\mathbb R^4\) has w.r.t. the canonical basis \(K\) analytic expression \(g(u)=2x^2+2xy-y^2-2yt-t^2\), where \(u=(x, y, z, t)^T\).

Find its analytic expression w.r.t. the basis

\(X=\{(1, 1, 1, 1)^T,(1, 1, 1, 0)^T,(1, 1, 0, 0)^T,(1, 0, 0, 0)^T\}\).

Determine \(g(u)\) for the vector \(u\), which has w.r.t. the basis \(X\) coordinates \([u]_X=(3, 1, 0, 0)^T\).

  • Hint

    Multiply the matrix \(B\) of the form \(g\) w.r.t. the canonical basis by the transition matrix from \(X\) to \(K\) from both sides to get its matrix w.r.t. the basis \(X\).

  • Resolution

    \( B_X=[id]_{XK}^T\cdot B_K \cdot [id]_{XK}\), i.e.

    \( B_X=\begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{pmatrix} \begin{pmatrix} 2 & 1 & 0 & 0 \\ 1 & -1 & 0 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & -1 \\ \end{pmatrix} \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{pmatrix} = \begin{pmatrix} 0 & 2 & 2 & 3 \\ 2 & 3 & 3 & 3 \\ 2 & 3 & 3 & 3 \\ 3 & 3 & 3 & 2 \\ \end{pmatrix} \)

    The analytic expression is: \( g(u)_X = 4ab + 4ac + 6ad + 3b^2 + 6bc + 6bd + 3c^2 + 6cd +2 d^2 \) for \([u]_X=(a, b, c, d)^T\).

    By substitution \((3, 1, 0, 0)^T\) for \((a, b, c, d)^T\) we get that \(g(u)=15\). The same result could be obtained if we calculate the coordinates of \(u\) w.r.t. \(K\), i.e. \([u]_K=(4, 4, 4, 3)^T\).

Difficulty level: Moderate task
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