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mathematics
linear algebra
Questions and Answers of
Linear Algebra
(a) Find all roots of f(x) = x2 + 4x if f(x) ∈ Z12[x]. (b) Find four distinct linear polynomials g(x), h(x), s(x), t(x) ∈ Z12[x] so that f(x) = g(x) h(x) = s(x)t(x). (c) Do the results in part
Determine whether or not each of the following polynomials irreducible over the given fields. If it is reducible, provide factorization into irreducible factors. (a) x2 + 3x - 1 over Q, R, C (b) x4 -
Let f(x), g(x) e ∈ [x] with f(x) = x3 + 2x2 + ax - b, g(x) = x3 + x2 - bx + a. Determine values for a, b so that the gcd of f(x), g(x) is a polynomial of degree 2.
For Example 17.9, determine which equivalence class contains each of the following: (a) x4 + x3 + x + 1 (b) x3 + x2 + l (c) x4 + x3 + x2 + 1
An outline for the proof of Theorem 17.11 follows.(a) Prove that the operations defined in part (a) of Theorem 17.11 are well-defined by showing that if f(x) = f1(x) (mods(x)) and g(x) = g1(x) (mod
(a) Show that s(x) = x2 + 1 is reducible in Z2[x]. (b) Find the equivalence classes for the ring Z2[x]/(s(x)). (c) Is Z2[x]/ (s(x)) an integral domain?
For the field in Example 17.11, find each of the following:(a) [x + 2] [2.x + 2] + [x + 1](b) [2x + l]2[x + 2](c) (22)-1 = [2x + 2]-1
Let s(x) = x4 + x3 + 1 ∈ Z2[x]. (a) Prove that s(x) is irreducible. (b) What is the order of the field Z2[x]/(s(x))? (c) Find [x2 + x + l]-1 in Z2[x]/(s(x)). (Find a, b, c, d ∈ Z2 so that [x2 + x
For p a prime, let be irreducible of degree n in Zp[x]. (a) How many elements are there in the field Zp [x]/(s(x))? (b) How many elements in Zp[x]/(s(x)) generate the multiplicative group of nonzero
For Theorem 17.13, prove that |S2| = P3.
Construct a finite field of 25 elements
Construct a finite field of 27 elements.
(a) Prove that the function h in Example 17.10 is one-to- one and onto and preserves the operation of addition.(b) Let (F, +, •) and (K, ⊕, ⊙) be two fields. If g: F → K is a ring isomorphism
(a) Let Q[√2] = (a + b√2|a, b ∈ Q}. Prove that (Q[[√2], +, •) is a subring of the field (R, +, •)• (Here the binary operations in R and Q[√2] are those of ordinary addition and
Let p be a prime, (a) How many monic quadratic (degree 2) polynomials x2 + bx + c in Zp[x] can we factor into linear factors in Zp[x]? (For example, if p = 5, then the polynomial x2 + 2x + 2 in
Let f(x) = (2x2 + 1) (5x3 - 5x + 3)(4x - 3) ∈ Z7[x]. Write f(x) as the product of a unit and three monic polynomials.
Prove Theorem 17.7. For polynomials in F[x], (a) Every nonzero polynomial of degree < 1 is irreducible. (b) if fix) ∈ F[x] with degree f(x) = 2 or 3, then f(x) is reducible if and only if f(x) has
An outline for a proof of Theorem 17.8 follows. (a) Let S = {s(x)f(x) + r(x)g(x)s(x), t(x) ∈ F[x]}. Select an element m(x) of minimum degree in S. (Recall that the zero polynomial has no degree, so
Prove Theorems 17.9 and 17.10.Euclidean Algorithm for Polynomials let fix), g(x) F[x] with degree f(x)Then rk(x), the last nonzero remainder, is a greatest common divisor of f(x), g(x),
Use the Euclidean algorithm for polynomials to find the gcd of each pair of polynomials, over the designated field F. Then write the gcd as s(x)f(x) +t(x)g(x) where s(x), t(x) ( F[x]. a) f(x) = x2 +
a. Rewrite the following 4 × 4 Latin square in standard form. 1 3 4 2 3 1 2 4 2 4 3 1 4 2 1 3 b. Find a 4 × 4 Latin squares in standard form that is orthogonal to the result
Prove Theorem 17.14. L1, L2 be an orthogonal pair of n X n Latin squares. If L1, L2 are standardized as L*1, L*2, then L*1, L*2 are orthogonal.
Complete the proof of the first part of Theorem 17.16. Let n ∈ Z+, n > 2. If p is a prime and n = p΄, for t ∈ Z+, then there are n - 1 Latin squares that are n × n and orthogonal in pairs.
The three 4 × 4 Latin squares in Tables 17.3, 17.4, and 17.6 are orthogonal in pairs. Can you find another 4 × 4 Latin square that is orthogonal to each of these three?
Complete the calculations in Example 17.17 in order to obtain the two 5 × 5 Latin squares L3 and L4. Rewrite each Latin square L1, for 1 < i < 4, in standard form.
Find three 7 × 7 Latin squares that are orthogonal in pairs. Rewrite these results in standard form.
A Latin square L is called self-orthogonal if L and its transpose Ltr form an orthogonal pair.a) Show that there is no 3 × 3 self-orthogonal Latin square.b) Give an example of a 4 × 4 Latin square
Complete the following table dealing with affine planes.
How many parallel classes does each of the affine planes in Exercise 1 determine? How many lines are in each class?
Construct the affine plane AP (Z3). Determine its parallel classes and the corresponding Latin squares for the classes of finite nonzero slope.
Repeat Exercise 3 with Z5 taking the place of Z3.
Determine each of the following lines. (a) The line in AP (Z7) that is parallel to y = 4x + 2 and contains (3, 6). (b) The line in AP (Z11) that is parallel to 2x + 3y + 4 = 0 and contains (10,
Suppose we try to construct an affine plane AP (Z6) as we did in this section.a) Determine which of the conditions (Al), (A2), and (A3) fail in this situation.b) Find how many lines contain a given
The following provides an outline for a proof of Theorem 17.18. (a) Consider a parallel class of lines given by y = mx + b, where m ∈ F, m ≠ 0. Show that each line in this class intersects each
Mrs. Mackey gave her computer science class a list of 28 problems and directed each student to write algorithms for the solutions of exactly seven of these problems. If each student did as instructed
Consider a (v, b, r, k, λ)-design on the set V of varieties, where |V| = v > 2. If x, y ∈ V, how many blocks in the design contain either x or y?
In a programming class Professor Madge has a total of n students, and she wants to assign teams of m students to each of p computer projects. If each student must be assigned to the same number of
a) If a projective plane has six lines through every point, how many points does this projective plane have in all? b) If there are 57 points in a projective plane, how many points lie on each line
In constructing the projective plane from AP (Z2) in Example 17.22, why didn't we want to include the point (0, 0, 0) in the set P'?
Determine the values of v, b, r, k, and λ for the balanced incomplete block design associated with the projective plane that arises from AP(F) for the following choices of F: (a) Z5 (b) Z7 (c)
(a) List the points and lines in AP(Z3). How many parallel classes are there for this finite geometry? What are the parameters for the associated balanced incomplete block design?(b) List the points
Complete the following table so that the parameters v, b, r, k, λ in any row may be possible for a balanced incomplete block design.
Is it possible to have a (v, b, r, k, λ)-design Where (a) b = 28, r = 4, k = 3? (b) v = 17, r = 8, k = 5?
Given a (v, b, r, k, λ)-design with b = v, prove that if v is even, then λ is even.
A (v, b, r, k, λ)-design is called a triple system if k = 3. When k = 3 and λ = 1, we call the design a Steiner triple system.(a) Prove that in every triple system, λ(v - 1) is even and λv(v - 1)
Verify that the following blocks constitute a Steiner triple system on nine varieties. 128 147 234 279 389 468 135 169 256 367 459 578
A projective plane is coordinatized with the elements of a field F. If this plane contains 91 lines, what are | F | and char (F)?
Let V = {x1, x2, . . . , xv] be the set of varieties and {B1, B2, . . ., Bb] the collection of blocks for a (v, b, r, k, λ)- design. We define the incidence matrix A for the design bya)
Given a (v, b, r, k, λ)-design based on the v varieties of V, replace each of the blocks Bt, for 1 < I < b, by its complement B1 = V - B1. Then the collection {B1, B2, ... , Bb} provides the blocks
(a) Let f(x) = anxn +∙∙∙∙∙∙∙∙+ a1x + a0 ∈ Z[x]. If r/s ∈ Q, with gcd(r, s) = land/(r/s) = 0, prove that s|an and r|a0.(b) Find the rational roots, if any exist, of the following
(a) For how many integers n, where 1 < n < 1000, can we factor f(x) = x2 + x - n into the product of two first degree factors in Z[x]?(b) Answer part (a) for f(x) = x2 + 2x - n.(c) Answer part
Verify that the polynomial f(x) = x4 + x3 + x + 1 is reducible over every field F (finite or infinite).
If p is a prime, prove that in Zp[x],
For any field F, let f(x) = xn + an-1xn-l + ∙∙∙∙∙∙∙∙ + a1x + a0 ∈ F[x]. lf r1,r2, . . ., rn are the roots of f(x), and r1 ∈ F for all 1 < i < n, prove that (a) -an-i - r1 + r2 +
(a) If a projective plane has 73 points, how many points lie on each line?(b) If each line in a projective plane passes through 10 points, how many lines are there in the projective plane?
Write each of the following in exponential form, for x, y ∈ R+. (a) √xy3 (b) 4√81x-5y3 (c) 53√8x9y-5
Approximate each of the following on the basis that (to four decimal places) log2 5 = 2.3219 and log2 7 = 2.8074. (a) log2 10 (b) log2 100 (c) log2 (7/5) (d) log2 175
Given that (to four decimal places) In 2 = 0.6931, In 3 = 1.0986, and In 5 = 1.6094, approximate each of the following. (a) log2 3 (b) log5 2 (c) log3 5
Determine the value of x in each of the following. (a) log10 2 + log10 5 = log10 x (b) log4 3 + log4 x = log4 7 - log4 5
Solve for x in each of the following. (a) log10 x + log10 6 = 1 (b) In x - ln(x - 1) = In 3 (c) log3(x2 + 4x + 4) - log3(2x - 5) = 2
Determine the value of x if log2 x = (l/3) [log2 3 - log2 5] + (2/3) log2 6 + log2 17
Let b be a fixed positive real number other than 1. If a, c ∈ R+, prove that alogbc = clogba.
Evaluate each of the following. (a) 125-4/3 (b) 0.0272/3 (c) (4/3)(l/8)-2/3
Determine each of the following. (a) (53/4)(513/4) (b) (73/5)/(718/5) (c) (51/2)(201/2)
In each of the following find the real number(s) x for which the equation is valid. (a) 53x2 = 55x+2 (b) 4x-1 = (1/2)4x-1
Write each of the following exponential equations as a logarithmic equation. (a) 27 = 128 (b) 1251/3 = 5 (c) 10-4 = 1/10,000 (d) 2a = b
Find each of the following logarithms. (a) log10 100 (b) log10 (l/1000) (c) log2 2048 (d) log2 (l/64) (e) log4 8 (f) log8 2 (g) log16 1 (h) log27 9
Solve for x in each of the following. (a) logx 243 = 5 (b) log3 x = -3 (c) log10 1000 = x (d) logx 32 = 5/2
Solve for x in each of the following. (a) logx 243 = 5 (b) log3 x = -3 (c) log10 1000 = x (d) logx 32 = 5/2 Discuss.
Prove part (2) of Theorem A1.2. Theorem A1.2 Let b, r, s ∈ R+ where b is fixed and other than 1. Then logb(r/s) = logb r - logb s.
Let b, r ∈ R+ where b is fixed and different from 1. (a) For all n ∈ Z+, prove that logb rn = n logb r. (b) Prove that logb r-n = (-n) logb r for all n ∈ Z+.
Find each of the following.(a) A + B(b) (A + B) + C(c) B + C(d) A + (B + C)(e) 2A(f) 2A + 3B(g) 2C + 3C(h) 5C(= (2 + 3)C)(i) 2B - 4C (= 2B + (-4)C)(j) A + 2B - 3C(k) 2(3B)(1) (2 3)BLet
Expand each of the following determinants across the specified row as well as down the specified column.(a)row 2 and column 3 (b) row 1 and column 2
Expand each of the following determinants across any row or down any column.(a)(b) (c)
(a) Evaluate each of the following 3 Ã 3 determinants.(i)(ii) (iii) (iv) (b) State a general result suggested by the answers in part (a).
(a) Evaluate each of the following 3 Ã 3 determinants.(i)(ii) (iii) (b) Let a, b, c, d, e, f, g, h, i R. If evaluate (i) (ii) (iii)
Let A = (aij)n×n and B = (bij)n×n be two matrices. When the matrix product AB is formed, as defined in Definition A2.5, how many multiplications (of entries) are performed? How many additions (of
Solve for a, b, c, d if
Perform the following matrix multiplications.(a)(b) (c) (d) (e) (f)
Show that(a) AB + AC = A(B + C); and(b) BA + CA = (B + C)A.[In general, if A is an m à n matrix and B, C are n à p matrices, then AB + AC = A(B + C). For n à p
Find the multiplicative inverse of each of the following matrices if the multiplicative inverse exists.(a)(b) (c) (d)
Solve each of the following matrix equations for the 2 Ã 2 matrix A.(a)(b)
Determine the following.(a) A-1(b) B-1(c) AB(d) (AB)-1(e) B-1A-1If
Solve the following systems of linear equations by using matrices: (a) 3x - 2y = 5 4x - 3y = 6 (b) 5x + 3y = 35 3x - 2y = 2
Determine whether each of the following statements is true or false. For parts (d)-(g) provide a counterexample if the statement is false. (a) The set Q+ is countable. (b) The set R+ is
(a) Let A = {n2|n ∈ Z+}. Find a one-to-one correspondence between Z+ and A. (b) Find a one-to-one correspondence between Z+ and {2, 6, 10, 14, ...}.
If S, T are infinite and countable, prove that S × T is countable.
Prove that Z+ × Z+ × Z+ = {(a, b, c) | a, b, c ∈ Z+} is countable.
Prove that the set of all real solutions of the quadratic equations ax2 + bx + c = 0, where a, b, c ∈ Z, a ≠ 0, is a countable set.
Determine a one-to-one correspondence between the open interval (0, 1) and the open intervals (a) (0, 3); (b) (2, 7); and (c) (a, b), where a, b ∈ R and a < b.
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