Math 240 Homework Assignment Three Do the following problems showing all your work. Be sure to submit your solutions in PDF or Word format to the Assignment area of LEO. (Point values are given in the square brackets.) 1. [4] Show (that is, verify all the axioms) that the set of second degree polynomials defined as, ! = = ! + + , , } is a vector space. What is the dimension of this vector space? 2. [4] Find the inverse of the linear transformation from ! ! . ! 22 8 13 3 ! 16 3 2 2 = ! + ! + ! + ! ! 8 9 7 2 ! 5 3 4 1 3. [4] Find the rotation matrix C that transforms this transformation. 0 3 . Make a diagram of 5 4 4. [3] Decide whether the matrix is invertible, if it is, find the inverse. 2 5 0 0 1 3 0 0 = 0 0 1 2 0 0 2 5 5. [3] Show that ! is a subspace of ! (See Problem #1 for the definition of the polynomial vector spaces.) 6. [3] Do the polynomials ! + 2 + 1, ! + 2, ! + 2, and ! + ! 5 + 2 span ! ? (Give support for your conclusion.) 7. [4] Let S be the set of vectors in ! ( = , , , , ) where, = = = = = , , , , , , , , , , , , , , , Find a subset of S that is a basis for the span(S)