6. [Bonus 4 points] Let V be the vector space of quadratic polynomials with real coefcients. It was shown in class that this vector space has dimension 3, with basis can. 1, and that % is a linear transformation on V. If we were working with taylor series multiplication by a.- would he too, but to get this to work on V we have to throw out any higher degree terms {so that :r - :2 is taken to be zero}. Do the following steps: a. Calculate a matrix for multiplication by I using the basis 1'2, 1:, 1 {he sure to use the same order as the example in class}. is. Describe the range and null space. c. If A denotes the matrix for i computed in class, and B denotes the matrix for multiplication by .1: computed in part a, show that AB EA is diagonal and not equal to zero. In quantum mechoncics corresponds to momentum and mattipii cation try .1: corresponds to position. In this may, the ooicuiotion in this port is o. partial proof of the Heisenberg uncertainty principie for position and momentum