1.8 let V be the space of all polynomials in one variable with real coefficients and of degree less than, or equal to, 3. Define the linear transformation T(ao+a1x+a2x +3x) = 00 +1(1+x) + a2(1+r) + 3(1 + x). Write down the matrix of T with respect to the basis {1, 1+x, 1x2, 1+x3}. 1.9 Let A be a 2 2 matrix with real entries which is not a diagonal matrix and which satisfies A3 = I. Pick out the true statements: a. tr(A) = -1; b. A is diagonalizable over R; c. = 1 is an eigenvalue of A. 1.10 Let A be a symmetric n n matrix with real entries, which is positive semi-definite, i.e. x Ax > 0 for every (column) vector x, where x denotes the (row) vector which is the transpose of x. Pick out the true statements: a. the eigenvalues of A are all non-negative; b. A is invertible; c. the principal minor Ak of A (i.e. the determinant of the k k matrix obtained from the first k rows and first k columns of A) is non-negative for each 1 kn.