Analytic expression for basis change
Task number: 2740
The quadratic form \(g\) on the vector space \(\mathbb R^4\) has w.r.t. the standard basis \(E\) analytic expression \(g(u)=2x^2-2xy+y^2-2yt+t^2\), where \(u=(x, y, z, t)^{\mathrm T}\).
Find its analytic expression \(g(u)_B\) w.r.t. the basis
\(B=\{(1, 1, 1, 1)^{\mathrm T},(0, 1, 1, 1)^{\mathrm T},(0, 0, 1, 1)^{\mathrm T},(0, 0, 0, 1)^{\mathrm T}\}\).
Determine \(g(u)\) for the vector \(u\), which has w.r.t. the basis \(B\) coordinates \([u]_B=(3, 1, 0, 0)^{\mathrm T}\).
Hint
Multiply the matrix \(A_E\) of the form \(g\) w.r.t. the standard basis by the transition matrix from \(B\) to \(E\) from both sides to get its matrix \(A_B\) w.r.t. the basis \(B\).
Solution
\( A_B=[id]_{BE}^{\mathrm T}\cdot A_E \cdot [id]_{BE}\), i.e.
\( A_B=\begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix} \begin{pmatrix} 2 & -1 & 0 & 0 \\ -1 & 1 & 0 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 1 \\ \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \\ \end{pmatrix} = \begin{pmatrix} 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \end{pmatrix} \)
The analytic expression is: \( g(u)_B = -2ab + c^2 + 2cd + d^2 \) for \([u]_B=(a, b, c, d)^{\mathrm T}\).
By substitution \((3, 1, 0, 0)^{\mathrm T}\) for \((a, b, c, d)^{\mathrm T}\) we get that \(g(u)=-6\). The same result could be obtained if we calculate the coordinates of \(u\) w.r.t. \(E\), i.e. \([u]_E=(3, 4, 4, 4)^{\mathrm T}\).
