1.3.2 Determine which of the following functions are and are not linear forms. *(a) The function f : R" → R defined by f(v) = ||v||. (b) The function f : F" → F defined by f(v) =v1. *(c) The function f: M2 → M2 defined by %3D | a b. a d (d) The determinant of a matrix (i.e., the function det : Mn → F). ẩjThe funcẩiẩnẩẩj: P → R defined by g(f) = f'(3), where f' is the derivative of f. (f) The function g :C → R defined by g(f) = cos(f(0)).

4. Let E: P3 R2 be given by E (p(x)) = [P(2)] (a) Find a matrix that induces the transformation E. TH (b) Find a polynomial p(z) where E(p(x)) = [], or explain why this isn't possible. (c) Find a polynomial q(2) where E(q(z)) = [], or explain why this isn't possible. 8P (d) Find a nonzero polynomial r(z) where E(r(2)) = [] or explain why this isn't possible. (e) Describe the kernel of E. What is the dimension of the kernel? Find a basis for the kernel. (f) Describe the image of E. What is the dimension of the image? Find a basis for the image. (g) Is the transformation E one to one? Is it onto? Is it an isomorphism? Explain.

(){88 B} (e) None of the above Problem 4. Let u = = and v are both solutions to Ax (a) det A = -1 (b) det A = 0 (c) det A = 1 (d) det A = 9 (e) None of the above 2 and b = 1 Let A be a 3 x 3 matrix such that u What is the determinant of A?