Coordinates in R4

In the space \(\mathbb R^4\) determine the coordinates \([u]_X\) with respect to the (ordered) basis

\(X=((1,-3{,}7,2)^T,(3{,}2,1,-4)^T,(0,-1{,}4,-3)^T,(-2{,}4,-3{,}0)^T)\)

for vectors \(u_1=(2{,}2,9,-5)^T,\ u_2=(-7{,}2,9,-8)^T\) a \(u_3=(4,-42{,}31,20)^T\).

Resolution
We solve three systems of equations that differ only in the right side. All could be together expressed by the matrix:

\( \left(\begin{array}{cccc|ccc} 1 & 3 & 0 &-2 & 2 &-7 & 4 \\ -3 & 2 &-1 & 4 & 2 & 2 &-42 \\ 7 & 1 & 4 &-3 & 9 & 9 & 31 \\ 2 &-4 &-3 & 0 &-5 &-8 & 20 \\ \end{array}\right) \sim\sim \left(\begin{array}{cccc|ccc} 1 & 0 & 0 & 0 & 1 & 0 & 2 \\ 0 & 1 & 0 & 0 & 1 &-1 & -4 \\ 0 & 0 & 1 & 0 & 1 & 4 & 0 \\ 0 & 0 & 0 & 1 & 1 & 2 & -7 \\ \end{array}\right) \)
Result
\([u_1]_X=(1{,}1,1{,}1)^T,[u_2]_X=(0,-1{,}4,2)^T,[u_3]_X=(2,-4{,}0,-7)^T\).