Chapter 4 - Vector Spaces - 4.7 Coordinates and Change of Basis - 4.7 Exercises - Page 210: 5

$$[x]_S=\left[ \begin {array}{ccc} 8\\ -3\end {array} \right].$$

Since we have $$x= 4(2,-1)+(0,1)=(8,-3).$$ then we can write $x$ relative to the standard basis of $R^2$ as follows $$x= (8,-3)=8(1,0)-3(0,1) .$$ Thus, the coordinates matrix of $x$ in $R^2$ relative to the standard basis is $$[x]_S=\left[ \begin {array}{ccc} 8\\ -3\end {array} \right].$$