## Change of a basis

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