Change of a basis
Task number: 2547
The coordinates of a vector \(u\) with respect to the ordered basis \(X=(v_1,v_2,v_3,v_4)\) are \([u]_X=(a_1,a_2,a_3,a_4)^T\). Determine the coordinates of the same vector with respect to the basis \(Y=(v_1+v_4,v_2+v_3,v_4,v_2)\).
Resolution
We seek \((b_1,…,b_4)^T=[u]_Y\), satisfying
\(b_1(v_1+v_4)+b_2(v_2+v_3)+b_3v_4+b_4v_2=a_1v_1+a_2v_2+a_3v_3+a_4v_4\).
Since \(X\) is a basis, the coefficients by \(v_i\) are unique. This yields a system
\( \begin{array}{rcl} b_1 & = & a_1 \\ b_2+b_4 & = & a_2 \\ b_2 & = & a_3 \\ b_1+b_3 & = & a_4 \end{array} \)
Result
The new coordinates are \([u]_Y=(a_1,a_3,a_4-a_1,a_2-a_3)^T\).
