12:39 1 .0 LTE Done 5 of 5 (3) Let V be the vector space of polynomials of degree at most n. Are the following subspaces? Prove or disprove. (a) All polynomials of the form f(x) = ar", where a is a real number. (b) All polynomials of the form f(x) = r" + a, where a is a real number. (c) All polynomials of degree at most n with integer coefficients. (d) All polynomials of degree at most n such that f (0) = 0. (4) Let V = R3. Define vector addition as 22 + " ]- [ X1 + yl [2 + 92 and scalar X3 + y3 O multiplication as c I 2 Show that all the axioms of a vector space are satisified except the last one, 1 . v = v. (5) Show that R2 is a subspace of R3. (Hint: Take R2 =" luck)) (6) Let A be an m x n matrix. Show that S = {x E R"|Ax = 0} is a subspace of R". (7) Let V be the vector space of polynomials with real coefficients with degree at most n. Show that S = {f E V| fo f(x)dx = 0} is a subspace of V. (8) Which of the following subsets of R2 are subspaces? Prove or disprove. (a) s = { }, where a 2 0 (b)s={[=]} (c) s = { } where k > 0. (d) s = {[mato] } for fixed real numbers m # 0 and b # 0. 2.2. INDEPENDENT SETS AND BASIS 51 (e ) S = (9) Let V be the vector space of 2 x 2 matrices with entries in R. Which of the following subsets of V are subspaces? Prove or disprove. (a) s= ab + )