Homework

Mathematics 115A, Lecture 1

Problem Set I, for Tuesday, April 8.
     Section 1.2, page 12, Exercises 11 and 19.   Also, a third one: Let V be the set of (strictly) positive real numbers.   Define the "sum" of vectors x and y to be the number xy (for example, the "sum" of vectors 2 and 3 is the vector 6). Define the "scalar product" of a number a and a vector x to be x^a (that is, x raised to the power a).   Is V, together with these two operations, a vector space over R?   Justify your answer.
     Section 1.3, page 19, Exercises 16, 23, and 28.   In #28, take n = 2 to simplify the notation.   You may use the properties of the transpose operation given in Exercise 3.   Cf. #5.   The field F is either R, C, or Q, because Z_2 has characteristic 2.
     Section 1.4, page 32, Exercises 10, 14, and 15.   In #10 and in #14, we need to show that two sets are equal; it might help to do the two inclusions one at a time.

(Solutions to Problem Set I have been distributed.)

Problem Set II, for Tuesday, April 15.

Section 1.5, page 40.

• Exercise 2, parts (b) and (d).
• Exercise 15.   (Assume that the n vectors are distinct.)   The point is to show that if a list of vectors is linearly dependent, then some vector is in the span of earlier vectors in the list.
• Exercise 18.   The set might be infinite, of course.
• Also: Assume that S1 and S2 are each linearly independent sets of vectors.
    (a) Show by example that the union of S1 and S2 might be linearly dependent.
    (b) Now suppose we add the assumption that span(S1) and span(S2) have only the zero vector in common.   Show that in this case the union of S1 and S2 is always linearly independent.

Section 1.6, page 53.

• Exercise 14.
• Exercise 18.   Example 5 is on page 11.
• Exercise 30.   Note the answers on page 572.   Give a basis for each of the four spaces.
• Exercise 33, part (a).

(Solutions to Problem Set II have been distributed.)

Problem Set III, for Tuesday, April 22.

Section 2.1, page 74.

• Exercise 9.
• Exercise 13.   It is to be understood that w1,w2, ... , wk are distinct.
• Exercise 14(b).
• Exercises 18 and 19 (count as one).

Section 2.2, page 84.

• Exercise 4.
• In Exercise 4, give the nullity and the rank of T, and find a basis for its kernel.
• Exercise 5(c,d).   Here tr(A) is the trace of A.
• In Exercise 5(c) and 5(d), give the nullity and the rank of T, and find a basis for its kernel.

(Solutions to Problem Set III have been distributed.)

Problem Set IV, for Tuesday, May 6.

Section 2.2, page 86.

• Exercise 14.

Section 2.3, page 96.

• Exercise 3(a).   The answers are on page 573.
• Exercise 12(a, b).   Give specific counterexamples where called for.   Part (c) is one of those things everyone needs to know; presumably you have seen it before.   Suggestion: See the 4/29/08 discussion section.
• Exercise 16(a).   Suggestion: What can you say about the restriction of T to its range?   You may use #29(a) from Section 1.6; see the 4/15/08 discussion section.

Section 2.4, page 106.

• Exercise 14.   There is more than one possible isomorphism from V onto F3.
• Exercise 16   Here B is fixed.
• Also: Assume that a and b are real numbers.   Define the linear operator T from P3(R) to P3(R) by the equation T(f) = af '' + bf ' + f.   Does it follow that T must be an isomorphism from P3(R) onto itself?   A counterexample or a proof would be appropriate.

Section 2.5, page 117.

• Exercise 4, but with T(a,b) = (2b - 5a, 5b - 12a) and with beta' = {(1, 2), (1, 3)}.   (The vectors written here as row vectors should actually be column vectors.)   (Be sure to use the method specified.   You know another method for finding [T] relative to beta', which can be used as a check.)
• Exercise 8.
• Exercise 10.   Exercise 13 on page 97 may be used without proof.   The significance of Exercise 10 is that for a linear operator T on a finite-dimensional space, we may speak unambiguously about the trace of T.

(Solutions to Problem Set IV have been distributed.)

Problem Set V, for Tuesday, May 13.

Section 4.4, page 237:

• Exercise 4(a).

Section 5.1, page 256:

• Exercise 3(d).
• Exercise 4(e).
• Exercise 14.
• Exercise 17(a).   Suggestion: What happens if we apply T twice?

Also:

• Assume that T is a linear transformation from V to W (where V and W are finite-dimensional vector spaces).   Assume that A is the matrix that represents T (with respect to some ordered bases for V and W).  
  (a) Show a vector v is in the kernel of T if and only if its coordinate vector [v] is in the nullspace of A.
  (b) Conclude that the kernel of T is isomorphic to the nullspace of A, under the coordinate map (restricted).
• Assume that gamma = {v1, ... , vn} is a list of linearly independent vectors in the n-dimensional space V (and hence a basis).   Assume that w1, ... , wn are any vectors in the m-dimensional vector space W.   Then there exists a unique linear transformation T from V to W for which T(vi) = wi for each i between 1 and n.   The point of this problem is to calculate T.  
  Let beta be our preferred ordered basis for V, and let delta be an ordered basis for W.   Let Q be the n x n matrix whose columns are [v1], ... , [vn], where the coordinates are with respect to beta.   Let Y be the m x n matrix that whose columns are [w1], ... , [wn], where the coordinates are with respect to delta.   Show that the matrix representing T with respect to beta and delta is (Y)x(the inverse of Q).

(Solutions to Problem Set V have been distributed.)

Problem Set VI, for Tuesday, May 20.

Section 5.2, page 279:

• Exercise 3(b).

Also:

• VI-A. In the space M(C) of 2 x 2 matrices over the complex field, determine whether the matrix B whose first row is [3   -4] and whose second row is [4   3] is similar to a diagonal matrix D. If so, give such a matrix D and a nonsingular matrix Q for which D = (Qinverse)(B)(Q).
• VI-B. For what value(s) of k, if any, is the 3 x 3 matrix A with rows
  [3   2   -3], [0   1   0], [2   k   -2] diagonalizable (over the real field)? For each such k, give a similar diagonal matrix D and a nonsingular matrix Q for which D = (Qinverse)(A)(Q).
• VI-C. Prove that every symmetric 2 x 2 matrix (over the real field) is diagonalizable.
• VI-D. Prove that two diagonalizable matrices are similar if and only if they have the same characteristic polynomial.
• VI-E. Assume that A and B are similar n x n matrices.
  (a) Show that for any scalar x, the matrix A + xI is similar to B + xI.
  (b) Show by example (with n = 2) that A + C is not always similar to B + C, where C is another n x n matrix.