Solution

We think of the polynomial as the vector [a,b,c] so all polynomials are . The fact that the polunomial passes through the point [0,1] means and similarly the fact that it passes through [1,2] means that so we have the equations 1 = a and 2 = a+b+c

The main point is to create the right equations. We have, by plugging in the points in , the equations:

If we think of this as the

Now as matrix equations, these are:

So the polynomials are the solutions to the equation AX =B which we can solve by elimination on the augmented system

Now, the system is rank 2 and hence the solution has one free variable. Thus it is a line full of solutions. Note, however, that the solutions do not form a vector space! They form a line not thru the origin. since is not a solution.