How to find the coordinates under a basis when variables are involved I understand how to use row operations when trying to find coordinates of a vector under a basis when both are involving numbers. Simply create an augmented matrix and row reduce. But how do I go about finding the coordinates of the vector ($2x+1+ \frac{x^3}{3} + \frac{x^2}{2}$) under the  basis B = ($x,x^2+x, 3-2x,x^3$)
I already have the solution, can someone please explain the steps to arrive there? 

Writing the vector as a combination of the base we have:
$$2x+1+ \frac{x^3}{3} + \frac{x^2}{2}=a(x)+b(x^2+x)+c(3-2x)+d(x^3)$$
$$2x+1+ \frac{x^3}{3} + \frac{x^2}{2}=dx^3+bx^2+(a+b-2c)x+3c$$
and using polynomial identity we get:
$$d=1/3$$
$$b=1/2$$
$$a+b-2c=2$$
$$3c=1$$
so 
$$a=13/6$$
$$b=1/2$$
$$c=1/3$$
$$d=1/3$$
