The required vector x is
.
Step-by-step explanation: Given that a basis B of R² consists of vectors (5, -6) and (-2, -2).
We are to find the vector x in R² whose co-ordinate vector relative to the basis B is
.
Let us consider that a, b are scalars such that
![a(5,-6)+b(-2,-2)=(-6,2)\\\\\Rightarrow (5a-2b,-6a-2b)=(-6,2)\\\\\Rightarrow 5a-2b=-6~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\\\\-6a-2b=2~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)](/tpl/images/0503/6217/2c332.png)
Subtracting equation (ii) from equation (i), we get
![(5a-2b)-(-6a-2b)=-6-2\\\\\Rightarrow 11a=-8\\\\\Rightarrow a=-\dfrac{8}{11}](/tpl/images/0503/6217/5af0f.png)
and from equation (i), we get
![5\times\left(-\dfrac{8}{11}\right)-2b=-6\\\\\\\Rightarrow 2b=-\dfrac{40}{11}+6\\\\\\\Rightarrow 2b=\dfrac{26}{11}\\\\\\\Rightarrow b=\dfrac{13}{11}.](/tpl/images/0503/6217/8595b.png)
Thus, the required vector x is
.