Homework #1

In what follows, we use the notation x^n to mean the n'th power of x. For instance, x^3 = xxx and x^4 = xxxx.

As discussed in class today, the order of an element x in a group G is the least integer n such that the n'th power of x is the identity: x^n = e.

#1 Let (Z, +) denote the group of integers Z under addition. Show that the subset dZ of Z consisting of the multiples of d is a subgroup of Z. In fact, more is true: Show that all subgroups of Z are of the form nZ for some integer n.

#2 Let a,b be elements of a group G. Assume that a has order 5 (i.e. aaaaa = e, the identity) and that a^3b = ba^3. Prove that ab = ba.

#3 Prove that in any group the orders of ab and ba are equal.

#4 Determine all the equivalence relations on a set of five elements.
 

Homework #2 (Due Sept 19)

#1 Describe all groups which have no proper subgroups.

#2 Prove that a group in which every element other than the identity has order 2 is abelian.

#3 Find all subgroups of S_3 =symmetric group on 3 letters, and determine which of them are normal.

#4 Let f: G --> G' be a homomorphism, and let x be an element of G of order r. What can you say about the order of the element f(x) of G' ?

#5 Let H and K be subgroups of a group G of orders 3 and 5 respectively. Prove that the intersection of H with K consists of just the identity element e of G.

#6 Determine all homomorphisms f: (Z, +) ---> (Z, +), and determine which of them are injective, which are surjective and which are isomorphisms.

#7 Let a be the element of S_3 which corresponds to the permutation 123 ---> 231 (we called this element alpha in class) and t be the element which corresponds to the permutation 123 ---> 213. So, for instance, t sends 321 to 231 (it just exchanges the first two numbers; we called this tau in class). We showed that S_3 = { e, a, aa, t, at, aat}. Let H be the subgroup {e, t} of S_3. Compute the product sets (eH).(aH) and (eH).(aaH), and verify that they are not cosets of H.
 

Homework #3 (Due Sept 26)

#1 Prove that every group of even order contains an element of order 2.

#2 Let a be an element of G and b be an element of G'. If n is the order of a and m is the order of , what is the order of the element (x,y) of G x G' (the product group) ?

#3 Classify all groups of order 6 by analyzing the following cases:

(i) G contains an element of order 6

(ii) G contains an element of order 3 but none of order 6

(iii) All elements of G have order 1 or 2.

#4 Prove that a group of order 30 can have atmost 7 distinct subgroups of order 5.

#5 Let G be a finite group whose order is the product of two integers: n = ab. Let H and K be subgroups of G of orders a and b respectively. Assume that H intersect K in just the identity, i.e., H and K have only the identity element in common. Prove that HK = G. In other words, every element of G is a product hk where h is in H and k is in K. Is G isomorphic to the product group H x K?
 

Homework #5 (Due October 10)

#1 Prove that the space M_n(R) of real n-by-n matrices is a vector space. A n-by-n matrix M is said to be symmetric if it is equal to its transpose M^t (i.e the matrix obtained by interchanging rows and columns) matrices (matrices which are equal to their transpose). It is said to be skew-symmetric if M = -M^t (the negative of the transpose). Let V be the subset of M_n(R) consisting of the symmetric matrices and W the subset consisting of the skew-symmetric matrices. Show that V and W are subspaces of M_n(R). Show that M_n(R) is the direct sum of V and W.

#2 What is the codimension of V in M_n(R)? (as in Problem #1).

#3 (i) Prove that the set B= (1,2,0), (2,1,2), (3,1,1) is a basis for 3-dimensional space R^3.

(ii) Find the coordinate vector of the vector (1,2,3) in this basis B.

(iii) Let B' = (0,1,0), (1,0,1), (2,1,0) be another basis (you dont have to check that this is a basis). Find the matrix which gives the change of basis from B to B'.

#4 Let W be a subspace of a vector space V.

(i) Show that there is a subspace U of V such that U + W = V and U intersect W is the zero vector.

(ii) Prove that there is no subspace U such that W intersects U in the zero vector and that dim W + dim U > dim V.
 

Practice Problems for Exam I

#1 Evaluate the determinant of the n x n matrix

--- ---

The first row has 123...n, the second 223...n, the third 3334...n, the fourth 44445...n, etc. until the last row which just consists of nnnnnnn.

#2 Let A be a m-by-m matrix. What is det(-A) in terms of det(A) ?

#3 Which of the following binary operations on the real numbers are commutative? Which are associative?

(i) * defined to be a*b = a + b + ab

(ii) * defined to be a*b = a - b

(iii) * defined to be a*b = (a+b)/5

#4 Let n be a positive integer greater than one. Let Z be the integers. Why is the ring Z/nZ of residues (i.e. remainders modulo n) not a group under multiplication?

#5 Find the order of every elements in the cyclic group C_8 (this group has 8 elements).

#6 Let x be an element of a group G. Let y be the inverse of x. Show that the order of x is equal to the order of y.

#7 If g is an element of a group G and g has order 2, what is the inverse of g?

#8 Prove that any group of even order contains an element of order 2.

#9 Determine whether the following sets are vector spaces over the field R of real numbers.

(i) The set of all pairs (x,y) where x >0 with the standard operations on R^2.

(ii) The set of all polynomials f(x) with real coefficients and satisfying f(1) = 0.

(iii) Let A be any n-by-n matrix with real entries. Let x and y be column vector of size n. It is clear that if Ax= 0, and Ay = 0, then A(x+y) = 0. Is the set of all column vectors v (of size n) such that Av =0 a vector space over R.

# 10 For which values of a do the following vectors form a linearly independent set in 3-dimensional space R^3?

# 11 Show that if {u,v,w,x} is a linearly independent set of vectors in a vector space V, then {u,v,w} is also linearly independent.

# 12 Describe geometrically all the subspaces of 3-dimensional vector space R^3.
 

Homework #6 (Due November 7)

#1 Prove that the inverse of a symmetric n-by-n matrix A is also symmetric.

#2 Let v_1, v_2, v_3,..., v_n be a subset of a vector space V. Prove that the map g: F^n ---> V defined by g(x_1, x_2, ...,x_n) = x_1v_1 + x_2v_2...+ x_nv_n is a linear transformation. Note that the x_i's are scalars and v_i's are vectors.

#3 Let L be the subspace of R^2 spanned by the vector (1,1). Let U be the subspace spanned by the vector (1,3). Find all linear transformations T: R^2 ---> R^2 which carry the subspace L into the subspace U.

#4 Let A be a m-by-n matrix. Consider the function T: F^n ---> F^m which sends a (column) vector x of F^n to the column vector Ax of F^m. Show that T is a linear transformation. Prove that the dimension of the subspace Ker(T) is atleast n-m.

#5 Let V_n be the vector space of polynomials p(x) with real coefficients of degree atmost n. Consider the operator D: V_n --->V_n defined by sending any polynomial p(x) to its derivative p'(x). For instance, D(2x^3 -7x) = 6x^2 -7. Prove that D is a linear transformation. The vectors {1,x,...x^n} forms a basis for V_n. Determine the matrix of D with respect to this basis.

#6 (i) We know that det(AB) = det(A). det(B) for any n-by-n matrices A and B. The entries of A and B can be in any field F. Show that the invertible n-by-n matrices form a group under multiplication. This group will be denoted GL(n;F). (ii) Consider the finite field Z/pZ of residue classes of integers modulo a prime p. For instance, Z/3Z= {0,1,2}. The number of n-by-n matrices over Z/pZ is p^{n^2}. For instance, if p=3 and n=5, we get 3 raised to the power 25 (it is not the same as squaring 3^5). Therefore, the order of the group Gl(n; Z/pZ) is finite. What is its order? Working out the case p=3 and n=2,3,4 can help you find the pattern.
 

Homework #7 (Due November 14)

#1, #2 Exercises #2 and #3 in Textbook on page 95.

#3 Consider the linear transformation T from R^2 to R^2 (two dimensional space) given by sending the vector (a,b) to (a + 2b, 3a+4b). Let A be the matrix of T in the standard basis B= {e_1, e_2} for R^2. Let A^i mean the product of the matrix A with itself i-times with A^0 = I (the 2-by-2 identity matrix). Given any polynomial P(x) with real coefficients, we can make sense of the matrix P(A). For instance, if P(x) = 1- x +7x^2, then P(A) will be the matrix I - A + 7 A^2. Here, 7A^2 means the you take the product of A with itself and multiply each entry of the result by the number 7. Can you find a polynomial P(x) such that P(A) is the 2-by-2 zero matrix?

#4 Consider the linear functional tr: M_2(R) ---> R which sends a 2-by-2 matrix to its trace. Namely, if N is a 2-by-2 matrix, then tr(N) = sum of the diagonal entries of N. We have the standard basis for M_2(R) given by e_1, e_2, e_3, e_4 where e_1 has the first coordinate 1 and the rest zero etc. These give us a basis f_1, f_2, f_3, f_4 for the dual space M_2(R)* = L(M_2(R), R). Express the linear functional tr as a linear combination of the vectors f_1, f_2, f_3, f_4.

#5 (#4 continued) If B is a 2-by-2 matrix, let us make a linear functional T_B: M_2(R) ---> R by T_B(N) = tr(BN). Namely, first compute the product of B and N and then apply trace to the result. Write T_B as a linear combination of the vectors f_1, f_2, f_3, f_4. For which matrix B is T_B = tr?

#6 (#5 continued) Consider the map g: M_2(R) ---> M_2(R)* given by g(B) = T_B. Is g a linear transformation? If so, is g one-to one? onto?
 

Practice problems for Exam II

#1 Let V_n be the vector space of polynomials P(x) with real coefficients of degree atmost n. Consider the linear transformation T: V_n ---> V_n given by T(P(x)) = P(3x-5). For instance, T(x^2)= (3x-5)^2 and T(a) = a for any constant a (corresponds to a P of degree zero). Write the matrix for T in the standard basis {1,x,x^2, x^3} for the vector space V_3.

#2. Let V_n be as in #1. Consider the transformation S: V_4 ---> V_3 defined by S(P(x)) = dP/dx = P' the derivative of P. Let T: V_3 ---> V_4 be the transformation T(Q(x)) = xQ(x) (multiplies the given polynomial by x -- so the degree of T(Q(x)) is one more than that of Q(x)--- which is why the range is V_4). Compute the matrix of ST: V_3 ---> V_3 in the standard basis for V_3. Is S one-to-one, onto? How about T?

#3. Let S and T be as in #2. The standard basis for V_3 gives us a basis E_{ij} for L(V_3, V_3). Express the vector ST of L(V_3, V_3) as a linear combination of the E_{ij}s.

#4. Consider R^4 (column vectors) with its standard basis B= {e_1, e_2, e_3, e_4}. Define a linear transformation N: R^4 ---R^4 by N(a,b,c,d)= (a+2b, c-5d, a+b_c+d, d+a). Compute the matrix of N in the standard basis B and the new basis B'= {f_1 = e_2,f_2 = e_3, f_3 = e_4, f_4 = e_1}.

#5. #1, #7, #8 from the Textbook on page 95.
 

Practice problems for Final Exam

First point: Since the exam is somewhat cumulative, certainly all the problems for the first two exams are still good practice problems. Also, problems from the handout (first section) page 262-3 are also good ones.

#1 Let u= (a,b) and v= (c,d) be vectors in R^2. Verify that the following is an inner product (i.e. it is bilinear, symmetric and positive definite): = 3ac + 4bd.

#2 What are the unit vectors in the inner product of #1 ? Can you draw it?

#3 Write the inner product in #1 as a matrix using the standard basis for R^2. What matrix do you get if you used the basis (1,2) and (3,7) instead?

#4 Let V_2 be the vector space of polynomials of degree 2 with real coefficients. Given p(x) = a + bx +cx^2 and q(x) = d + ex + fx^2, two vectors of V_2, define (p,q) = 9 ad +be + cf. Check that this is a symmetric bilinear form. Is it positive definite? If so, then we have an inner product on V_2. Do you know of an orthonormal basis for V_2 with respect to this inner product?

#5 Consider the bilinear form ( , ) defined on R^2 by the following matrix A (with respect to the standard basis). A= (a_{ij}) where a_11 = 3 = (e_1, e_1), a_12 = (e_1, e_2) = -2, a_21 = (e_2, e_1) = 4, and a_22 = (e_2, e_2) =8. Compute the number (u,v) where u = [3,-2] and v= [ -1,6].

#7 Use the bilinear form of #5 on R^2. Can you find two vectors v and w of R^2 such that v is orthogonal to w but w is not orthogonal to v?

#8 Find an orthonormal basis for R^2 using the bilinear form in #1.

#9 Let R^3 have the usual dot product as the inner product. Use the Gram-Schmidt procedure to convert the following basis {v_1, v_2, v_3} into an orthornormal basis for R^3.

v_1 = [1,1,1], v_2 = [-1,1,0] and v_3 = [1,2,1].

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