Question

b-c a (a) (5 points) Find a matrix representation for this linear transformation. (b) (3 points) What is the nullspace of this matrix? Write it as a span (e) (3 points) What is the column space of this matrix? Write it as a span.

Answer 1

F(1) = (0 1 0 0) = e_12 f(x) = (0 0 -1 0) = -e_21 f(x^2) = (0 0 1 0) = e_21 f(x^3) = (0 1 0 1) = e_12 + e_22 therefore matrix = [0 0 0 0 1 0 0 0 0 -1 1 0 0 0 0 1) let ax^3 + cx^2 + cx + alpha ker (f) + u_x f(ax^3 + bx^2 + cx + d) = 0 a + d = 0 b - c = 0, 0 = 0 a = 0 implies d = 0 b = c therefore ax^3 + bx^2 + cx + d = 6(x^2 + x) therefore null space = span {x^2 + x} note that 3rd column is multiple of second column therefore column space = span { (0 1 0 0) , (0 0 1 0), (0 1 0 1)}