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


    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 \([u]_Y=(a_1,a_3,a_4-a_1,a_2-a_3)^T\).

Difficulty level: Easy task (using definitions and simple reasoning)
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