I came across the following question:
Let be the real vector space of polynomials with degree . For let and . describes the derivative of .
Find a basis for and .
Unfortunatly I am not quite getting my head wrapped around vector spaces with polynomials yet. So after failing the question I had a look at the solution and there it says:
By solving the system of linear equations:
More explanation is not given.
I know that the elements in in the form of are zero at , so .
Respectively for the elements in :.
However, it is not clear to me how I get from that information to the basis of and using systems of linear equations.