Advanced Calculus fi..

70 Advanced Calculus, Fifth EditionEXAMPLE 4 The set of all complex numbers forms a complex vector space,whose dimension is 1, with a basis consisting of any one nonzero complex number.EXAMPLE 5 The set of all polynomials with complex coefficients forms aninfinite-dimensional complex vector space.EXAMPLE 6 The set of all functions of form ae2'-' + be-"" , where a and bare complex constants, forms a complex vector space of dimension 2, with basise2i.t p-2i.rPROBLEMS1. Show that each of the following sets of objects, with the usual operations of addition andmultiplication by scalars, forms a vector space. Give the dimension in each case and, ifthe dimension is finite, give a basis.a) All polynomials of degree at most 2.b) All polynomials containing no term of odd degree: 3 + 5x2 + x4, x2 - xI0, . . . .c) All trigonometric polynomials:a" + a1 cos x + bl sin x + . . . + a, cos nx + b, sin nx.d) All functions of the form aer + be-'.e) All 3 x 3 diagonal matrices.f) All 4 x 4 symmetric matrices A; that is, all matrices A such that A = A'.g) All functions y = f (x), -ca < x < ca, such that y" + = 0.h) All functions y = f (x). -CX < x < ca, such that y"' - q' = 0.i) All functions f (x) which are defined and continuous for 0 5 x 5 1.j) All functions f (x) which are defined and have a continuous derivative for 0 5 x 5 1.k) All infinite sequences: X I. .r,. . . . , x,, . . . .I) All convergent sequences.2. Show that the four polynomials 1, I + x. 1 + x + x'. 1 + x + x' form a basis for the vectorspacc of Example 2.3. Show that thc following matrices form a basis for the vector spacc of Example 3:4. Show that the functions cos2x, sin2x form a basis for the complex vector space ofExamplc 6.Suggested ReferencesBirkhoff, G., and S. MacLane, Survey of Modern Algebra. 4th ed. New York: Macmillan,1977.Cullen, Charles G., Matrices and Linear Transformation, 2nd ed. Reading, Mass.: Addison-Wesley, 1972.Curtis, Charles W., Linear Algebra: An Introductory Approach. New York: Springer-Verlag,1999.