Math 340 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, P, = {p(x)| p(x) = ax? + bx + c where a,b,c R} is a vector space. What is the dimension of this vector space? 2. [4] Find the inverse of the linear transformation from R? to R3, X1 2 3 1 T xZ = xl 3+ x2 3+ X3 1 X3 2 4 1 3. [4] Find the rotation matrix C that transforms [ 103] into [ 152]. Make a diagram of this transformation. 4. [3] Decide whether the matrix is invertible, if it is, find the inverse. 5 2 0 0 121 0 0 A'oozs 0 0 1 3 5. [3] Show that P, is a subspace of P; (See Problem #1 for the definition of the polynomial vector spaces.) 6. [3] Do the polynomials t3 +t + 2, t> t + 2,t> + 2, and t3 + t? 5t + 3 span P;? (Give support for your conclusion.) 7. [4] Let S be the set of vectors in R* (S = {vq, v3, V3, 4, V5 }) where, v1=(1,2,-2,-1) v, =(-3,0,4,3) ;3 =(2,1,-1,-1) vy, =(-3,3,-9,6) vs =(3,9,7,6) Find a subset of that is a basis for the span(S)