Mathematical Methods in the Physical Sciences
3rd Edition
ISBN: 9780471198260
Author: Mary L. Boas
Publisher: Wiley, John & Sons, Incorporated
expand_more
expand_more
format_list_bulleted
Textbook Question
Chapter 12.8, Problem 3P
Find the norm of each of the following functions on the given interval and state the normalized function.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
5. (a) State the Residue Theorem. Your answer should include all the conditions required
for the theorem to hold.
(4 marks)
(b) Let y be the square contour with vertices at -3, -3i, 3 and 3i, described in the
anti-clockwise direction. Evaluate
に
dz.
You must check all of the conditions of any results that you use.
(5 marks)
(c) Evaluate
L
You must check all of the conditions of any results that you use.
ཙ
x sin(Tx)
x²+2x+5
da.
(11 marks)
3. (a) Lety: [a, b] C be a contour. Let L(y) denote the length of y. Give a formula
for L(y).
(1 mark)
(b) Let UCC be open. Let f: U→C be continuous. Let y: [a,b] → U be a
contour. Suppose there exists a finite real number M such that |f(z)| < M for
all z in the image of y. Prove that
<
||, f(z)dz| ≤ ML(y).
(3 marks)
(c) State and prove Liouville's theorem. You may use Cauchy's integral formula without
proof.
(d) Let R0. Let w € C. Let
(10 marks)
U = { z Є C : | z − w| < R} .
Let f UC be a holomorphic function such that
0 < |ƒ(w)| < |f(z)|
for all z Є U. Show, using the local maximum modulus principle, that f is constant.
(6 marks)
3. (a) Let A be an algebra. Define the notion of an A-module M. When is a module M
a simple module?
(b) State and prove Schur's Lemma for simple modules.
(c) Let AM(K) and M = K" the natural A-module.
(i) Show that M is a simple K-module.
(ii) Prove that if ƒ € Endд(M) then ƒ can be written as f(m) = am, where a
is a matrix in the centre of M, (K).
[Recall that the centre, Z(M,(K)) == {a Mn(K) | ab
M,,(K)}.]
= ba for all bЄ
(iii) Explain briefly why this means End₁(M) K, assuming that Z(M,,(K))~
K as K-algebras.
Is this consistent with Schur's lemma?
Chapter 12 Solutions
Mathematical Methods in the Physical Sciences
Ch. 12.1 - Solve the following differential equations by...Ch. 12.1 - Solve the following differential equations by...Ch. 12.1 - Solve the following differential equations by...Ch. 12.1 - Solve the following differential equations by...Ch. 12.1 - Solve the following differential equations by...Ch. 12.1 - Solve the following differential equations by...Ch. 12.1 - Solve the following differential equations by...Ch. 12.1 - Solve the following differential equations by...Ch. 12.1 - Solve the following differential equations by...Ch. 12.1 - Solve the following differential equations by...
Ch. 12.2 - Using (2.6) and (2.7) and the requirement that...Ch. 12.2 - Show that Pl(1)=(1)l. Hint: When is Pl(x) an even...Ch. 12.2 - Computer plot graphs of Pl(x) for l=0,1,2,3,4, and...Ch. 12.2 - Use the method of reduction of order [Chapter 8,...Ch. 12.3 - By Leibniz' rule, write the formula for...Ch. 12.3 - Use Problem 1 to find the following derivatives....Ch. 12.3 - Use Problem 1 to find the following derivatives....Ch. 12.3 - Use Problem 1 to find the following derivatives....Ch. 12.3 - Use Problem 1 to find the following derivatives....Ch. 12.3 - Verify Problem 1. Hints: One method is to use...Ch. 12.4 - Verify equations (4.4) and (4.5). (4.4)...Ch. 12.4 - Show that Pl(1)=1, with P1(x) given by (4.1), in...Ch. 12.4 - Find P0(x),P1(x),P2(x),P3(x), and P4(x) from...Ch. 12.4 - Show that 11xmPl(x)dx=0 if ml. Hint: Use...Ch. 12.5 - Find P3(x) by getting one more term in the...Ch. 12.5 - Verify (5.5) using (5.1). (5.1)...Ch. 12.5 - Use the recursion relation (5.8a) and the values...Ch. 12.5 - Show from (5.1) that (xh)x=hh. Substitute the...Ch. 12.5 - Differentiate the recursion relation (5.8a) and...Ch. 12.5 - From (5.8b) and (5.8c), obtain (5.8d) and (5.8f)....Ch. 12.5 - Write (5.8c) with l replaced by l+1 and use it to...Ch. 12.5 - Express each of the following polynomials as...Ch. 12.5 - Express each of the following polynomials as...Ch. 12.5 - Express each of the following polynomials as...Ch. 12.5 - Express each of the following polynomials as...Ch. 12.5 - Express each of the following polynomials as...Ch. 12.5 - Express each of the following polynomials as...Ch. 12.5 - Show that any polynomial of degree n can be...Ch. 12.5 - Expand the potential V=K/d in (5.11) in the...Ch. 12.6 - Show that if abA*(x)B(x)dx=0 [see (6.3)], then...Ch. 12.6 - Show that the functions einx/l,n=0,1,2,, are a set...Ch. 12.6 - Show that the functions x2 and sinx are orthogonal...Ch. 12.6 - Show that the functions f(x) and g(x) are...Ch. 12.6 - Evaluate 11P0(x)P2(x)dx to show that these...Ch. 12.6 - Show in two ways that Pl(x) and Pl(x) are...Ch. 12.6 - Show that the set of functions sinnx is not a...Ch. 12.6 - Show that the functions cosn+12x,n=0,1,2,, are...Ch. 12.6 - Show in two ways that 11P2n+1(x)dx=0.Ch. 12.7 - By a method similar to that we used to show that...Ch. 12.7 - Following the method in (7.2) to (7.5), show that...Ch. 12.7 - Use Problem 4.4 to show that 11Pm(x)Pl(x)dx=0 if...Ch. 12.7 - Use equation (7.6) to show that 11Pl(x)Pl1(x)dx=0....Ch. 12.7 - Show that 11Pl(x)dx=0,l0. Hint: Consider...Ch. 12.7 - Show that P1(x) is orthogonal to Pl(x)2 on (1,1)....Ch. 12.8 - Find the norm of each of the following functions...Ch. 12.8 - Find the norm of each of the following functions...Ch. 12.8 - Find the norm of each of the following functions...Ch. 12.8 - Find the norm of each of the following functions...Ch. 12.8 - Find the norm of each of the following functions...Ch. 12.8 - Give another proof of (8.1) as follows. Multiply...Ch. 12.8 - Using (8.1), write the first four normalized...Ch. 12.9 - Expand the following functions in Legendre series....Ch. 12.9 - Expand the following functions in Legendre series....Ch. 12.9 - Expand the following functions in Legendre series....Ch. 12.9 - Expand the following functions in Legendre series....Ch. 12.9 - Expand the following functions in Legendre series....Ch. 12.9 - Expand the following functions in Legendre series....Ch. 12.9 - Expand the following functions in Legendre series....Ch. 12.9 - Expand the following functions in Legendre series....Ch. 12.9 - Expand the following functions in Legendre series....Ch. 12.9 - Expand each of the following polynomials in a...Ch. 12.9 - Expand each of the following polynomials in a...Ch. 12.9 - Expand each of the following polynomials in a...Ch. 12.9 - Find the best (in the least squares sense)...Ch. 12.9 - Find the best (in the least squares sense)...Ch. 12.9 - Find the best (in the least squares sense)...Ch. 12.9 - Prove the least squares approximation property of...Ch. 12.10 - Verify equations (10.3) and (10.4). (10.4)...Ch. 12.10 - The equation for the associated Legendre functions...Ch. 12.10 - Show that the functions Plm(x) for each m are a...Ch. 12.10 - Substitute the Pl(x) you found in Problems 4.3 or...Ch. 12.10 - Substitute the Pl(x) you found in Problems 4.3 or...Ch. 12.10 - Substitute the P1(x) you found in Problems 4.3 or...Ch. 12.10 - Show that...Ch. 12.10 - Write (10.7) with m replaced by m; then use...Ch. 12.10 - Use Problem 7 to show that...Ch. 12.10 - Derive (10.8) as follows: Multiply together the...Ch. 12.11 - Finish the solution of equation (11.2) when s=2....Ch. 12.11 - Solve the following differential equations by the...Ch. 12.11 - Solve the following differential equations by the...Ch. 12.11 - Solve the following differential equations by the...Ch. 12.11 - Solve the following differential equations by the...Ch. 12.11 - Solve the following differential equations by the...Ch. 12.11 - Solve the following differential equations by the...Ch. 12.11 - Solve the following differential equations by the...Ch. 12.11 - Solve the following differential equations by the...Ch. 12.11 - Solve the following differential equations by the...Ch. 12.11 - Solve the following differential equations by the...Ch. 12.11 - Solve the following differential equations by the...Ch. 12.11 - Consider each of the following problems as...Ch. 12.11 - Solve y=y by the Frobenius method. You should find...Ch. 12.12 - Show by the ratio test that the infinite series...Ch. 12.12 - Use (12.9) to show that: J2(x)=(2/x)J1(x)J0(x)Ch. 12.12 - Use (12.9) to show that: J1(x)+J3(x)=(4/x)J2(x)Ch. 12.12 - Use (12.9) to show that: (d/dx)J0(x)=J1(x)Ch. 12.12 - Use (12.9) to show that: (d/dx)xJ1(x)=xJ0(x)Ch. 12.12 - Use (12.9) to show that: J0(x)J2(x)=2(d/dx)J1(x)Ch. 12.12 - Use (12.9) to show that: limx0J1(x)/x=12Ch. 12.12 - Use (12.9) to show that: limx0x3/2J3/2(x)=312/...Ch. 12.12 - Use (12.9) to show that: x/2J1/2(x)=sinxCh. 12.13 - Using equations (12.9) and (13.1), write out the...Ch. 12.13 - Show that, in general for integral...Ch. 12.13 - Use equations (12.9) and (13.1) to show that:...Ch. 12.13 - Use equations (12.9) and (13.1) to show that:...Ch. 12.13 - Use equations (12.9) and (13.1) to show that:...Ch. 12.13 - Use equations (12.9) and (13.1) to show that: Show...Ch. 12.14 - By computer, plot graphs of Jp(x) for p=0,1,2,3,...Ch. 12.14 - From the graphs in Problem 1, read approximate...Ch. 12.14 - By computer, plot N0(x) for x from 0 to 15, and...Ch. 12.14 - From the graphs in Problem 3, read approximate...Ch. 12.14 - By computer, plot xJ1/2(x) for x from 0 to 4. Do...Ch. 12.14 - By computer, find 30 zeros of J0 and note that the...Ch. 12.15 - Prove equation (15.2) by a method similar to the...Ch. 12.15 - Solve equations (15.1) and (15.2) for Jp+1(x) and...Ch. 12.15 - Carry out the differentiation in equations (15.1)...Ch. 12.15 - Use equations (15.1) to (15.5) to do Problems 12.2...Ch. 12.15 - Using equations (15.4) and (15.5), show that...Ch. 12.15 - As in Problem 5, show that Jp1(x)=Jp+1(x) at every...Ch. 12.15 - (a) Using (15.2), show that 0J1(x)dx=J0(x)0=1. (b)...Ch. 12.15 - From equation (15.4), show that...Ch. 12.15 - Use L23 and L32 of the Laplace Transform Table...Ch. 12.16 - Find the solutions of the following differential...Ch. 12.16 - Find the solutions of the following differential...Ch. 12.16 - Find the solutions of the following differential...Ch. 12.16 - Find the solutions of the following differential...Ch. 12.16 - Find the solutions of the following differential...Ch. 12.16 - Find the solutions of the following differential...Ch. 12.16 - Find the solutions of the following differential...Ch. 12.16 - Find the solutions of the following differential...Ch. 12.16 - Find the solutions of the following differential...Ch. 12.16 - Find the solutions of the following differential...Ch. 12.16 - Find the solutions of the following differential...Ch. 12.16 - Find the solutions of the following differential...Ch. 12.16 - Verify by direct substitution that the text...Ch. 12.16 - Use (16.5) to write the solutions of the following...Ch. 12.16 - Use ( 16.5 ) to write the solutions of the...Ch. 12.16 - Use (16.5) to write the solutions of the following...Ch. 12.16 - Use (16.5) to write the solutions of the following...Ch. 12.17 - Write the solutions of Problem 16.1 as spherical...Ch. 12.17 - From Problem (12.9) J1/2(x)=2/xsinx. Use (15.2) to...Ch. 12.17 - From Problems 13.3 and 13.5, Y1/2(x)=2/x cos x. As...Ch. 12.17 - Using (17.3) and the results stated in Problems 2...Ch. 12.17 - Show from (17.4) that hn(1)(x)=ixn1xddxneixx.Ch. 12.17 - Using (16.1) and (17.4) show that the spherical...Ch. 12.17 - (a) Solve the differential equation xy=y using...Ch. 12.17 - Using (16.1) and (16.2), verify that (a) the...Ch. 12.17 - Using (17.3) and (15.1) to (15.5), find the...Ch. 12.17 - Computer plot (a) I0(x),I1(x),I2(x), from x=0 to...Ch. 12.17 - From (17.4), show that hn(1)(ix)=ex/x.Ch. 12.17 - Use the Section 15 recursion relations and (17.4)...Ch. 12.17 - Use the Section 15 recursion relations and (17.4)...Ch. 12.17 - Use the Section 15 recursion relations and (17.4)...Ch. 12.17 - Use the Section 15 recursion relations and (17.4)...Ch. 12.17 - Use the Section 15 recursion relations and (17.4)...Ch. 12.18 - Verify equation (18.3) Hint: From equation (18.2),...Ch. 12.18 - Solve equation (18.3) to get equation (18.4).Ch. 12.18 - Prove Jp(x)Jp(x)Jp(x)Jp(x)=2xsinp as follows:...Ch. 12.18 - Using equation (13.3) and Problem 3, show that...Ch. 12.18 - Use the recursion relations of Section 15 (for N s...Ch. 12.18 - For the initial conditions =0,=0, show that the...Ch. 12.18 - Prob. 7PCh. 12.18 - Find =ddt=ddududldldt either from equations...Ch. 12.18 - Consider the shortening pendulum problem. Follow...Ch. 12.18 - The differential equation for transverse...Ch. 12.18 - A straight wire clamped vertically at its lower...Ch. 12.19 - Prove equation (19.10) in the following way. First...Ch. 12.19 - Given that J3/2(x)=2xsinxxcosx, use (19.10) to...Ch. 12.19 - Use (17.4) and (19.10) to write the orthogonality...Ch. 12.19 - Define Jp(z) for complex z by the power series...Ch. 12.19 - We obtained (19.10) for Jp(x),p0. It is, however,...Ch. 12.19 - By Problem 5,01xN1/2(x)N1/2(x)dx=0 if and are...Ch. 12.20 - Use the table above to evaluate the following...Ch. 12.20 - Use the table above to evaluate the following...Ch. 12.20 - Use the table above to evaluate the following...Ch. 12.20 - Use the table above to evaluate the following...Ch. 12.20 - Use the table above to evaluate the following...Ch. 12.20 - Use the table above to evaluate the following...Ch. 12.20 - Use the table above and the definitions in Section...Ch. 12.20 - Use the table above and the definitions in Section...Ch. 12.20 - Use the table above and the definitions in Section...Ch. 12.20 - Use the table above and the definitions in Section...Ch. 12.20 - To study the approximations in the table, computer...Ch. 12.20 - To study the approximations in the table, computer...Ch. 12.20 - To study the approximations in the table, computer...Ch. 12.20 - To study the approximations in the table, computer...Ch. 12.20 - To study the approximations in the table, computer...Ch. 12.20 - To study the approximations in the table, computer...Ch. 12.20 - To study the approximations in the table, computer...Ch. 12.20 - To study the approximations in the table, computer...Ch. 12.20 - Computer plot on the same axes several Ip(x)...Ch. 12.20 - As in Problem 19, study the Kp(x) functions. It is...Ch. 12.21 - For Problems 1 to 4, find one (simple) solution of...Ch. 12.21 - For Problems 1 to 4, find one (simple) solution of...Ch. 12.21 - For Problems 1 to 4, find one (simple) solution of...Ch. 12.21 - For Problems 1 to 4, find one (simple) solution of...Ch. 12.21 - Solve the differential equations in Problems 5 to...Ch. 12.21 - Solve the differential equations in Problems 5 to...Ch. 12.21 - Solve the differential equations in Problems 5 to...Ch. 12.21 - Solve the differential equations in Problems 5 to...Ch. 12.21 - Solve the differential equations in Problems 5 to...Ch. 12.21 - Solve the differential equations in Problems 5 to...Ch. 12.21 - For the differential equation in Problem 2, verify...Ch. 12.21 - Verify that the differential equation x4y+y=0 is...Ch. 12.21 - Verify that the the differential equation in...Ch. 12.22 - Verify equations (22.2), (22.3), (22.4), and...Ch. 12.22 - Solve (22.9) to get (22.10). If needed, see...Ch. 12.22 - Show that ex2/2Dex2/2f(x)=(Dx)f(x). Now set...Ch. 12.22 - Using (22.12) find the Hermite polynomials given...Ch. 12.22 - By power series, solve the Hermite differential...Ch. 12.22 - Substitute yn=ex2/2Hn(x) into (22.1) to show that...Ch. 12.22 - Prove that the functions Hn(x) are orthogonal on...Ch. 12.22 - In the generating function (22.16), expand the...Ch. 12.22 - Use the generating function to prove the recursion...Ch. 12.22 - Evaluate the normalization integral in (22.15)....Ch. 12.22 - Show that we have solved the following eigenvalue...Ch. 12.22 - Using Leibniz' rule (Section 3), carry out the...Ch. 12.22 - Using (22.19) verify (22.20) and also find L3(x)...Ch. 12.22 - Show that y=Ln(x) given in ( 22.18 ) satisfies (...Ch. 12.22 - Solve the Laguerre differential equation...Ch. 12.22 - Prove that the functions Ln(x) are orthogonal on...Ch. 12.22 - In (22.23), write the series for the exponential...Ch. 12.22 - Verify the recursion relations (22,24) as follows:...Ch. 12.22 - Evaluate the normalization integral in (22.22)....Ch. 12.22 - Using (22.25),(22.20), and Problem 13, find Lnk(x)...Ch. 12.22 - Verify that the polynomials Lnk(x) in ( 22.25 )...Ch. 12.22 - Verify that the polynomials given by (22.27) are...Ch. 12.22 - Verify the recursion relation relations (22.28) as...Ch. 12.22 - Show that the functions Lnk(x) are orthogonal on...Ch. 12.22 - Evaluate the normalization integrals ( 22.29 ) and...Ch. 12.22 - Solve the following eigenvalue problem (see end of...Ch. 12.22 - The functions which are of interest in the theory...Ch. 12.22 - Repeat Problem 27 for l=0,n=1,2,3.Ch. 12.22 - Show that Rp=pxD and Lp=px+D where D=d/dx, are...Ch. 12.22 - Find raising and lowering operators (see Problem...Ch. 12.23 - Use the generating function (5.1) to find the...Ch. 12.23 - Use the generating function to show that...Ch. 12.23 - Use (5.78e) to show that...Ch. 12.23 - Obtain the binomial coefficient result in Problem...Ch. 12.23 - Show that 0n(2l+1)Pl(x)=Pn(x)+Pn+1(x). Hint: Use...Ch. 12.23 - Using (10.6), (5.8), and Problem 2, evaluate...Ch. 12.23 - Show that, for l0,0bP(x)dx=0 if a and b are any...Ch. 12.23 - Show that (2l+1)x21Pl(x)=l(l+1)Pl+1(x)Pl1(x)....Ch. 12.23 - Evaluate 11xPi(x)Pn(x)dx,nl. Hint: Write (5.8a)...Ch. 12.23 - Use the recursion relations of Section 15 (and, as...Ch. 12.23 - Use the recursion relations of Section 15 (and, as...Ch. 12.23 - Use the recursion relations of Section 15 (and, as...Ch. 12.23 - Wre the recursion relations of Section 15 (and, as...Ch. 12.23 - Use the recursion relations of Section 15 (and, as...Ch. 12.23 - Use the result of Problem 18.4 and equations...Ch. 12.23 - Use (15.2) repeatedly to show that...Ch. 12.23 - Let be the first positive zero of J1(x) and let n...Ch. 12.23 - (a) Make the change of variables z=ex in the...Ch. 12.23 - (a) The generating function for Bessel functions...Ch. 12.23 - In the generating function equation of Problem 19,...Ch. 12.23 - In the generating function equation, Problem 19,...Ch. 12.23 - In the cos(xsin) series of Problem 20, let =0, and...Ch. 12.23 - Solve by power series 1x2yxy+n2y=0. The polynomial...Ch. 12.23 - (a) The following differential equation is often...Ch. 12.23 - In Problem 22.26, replace x by x/n in the y...Ch. 12.23 - Verify Bauers formula eixw=0(2l+1)iiji(x)Pl(w) as...Ch. 12.23 - Show that R=lx1x2D and L=lx+1x2D, where D=d/dx,...Ch. 12.23 - Show that the functions J0(t) and J0(t) are...Ch. 12.23 - Show that the Fourier cosine transform (Chapter 7,...Ch. 12.23 - Use the results of Chapter 7, Problems 12.18 and...
Additional Math Textbook Solutions
Find more solutions based on key concepts
Absolute maximum/minimum values Use the following graphs to identify the points (if any) on the interval [a, b]...
Calculus: Early Transcendentals (2nd Edition)
Find how many SDs above the mean price would be predicted to cost.
Intro Stats, Books a la Carte Edition (5th Edition)
Assessment 1-1A The following is a magic square all rows, columns, and diagonals sum to the same number. Find t...
A Problem Solving Approach To Mathematics For Elementary School Teachers (13th Edition)
2. Quantitative Concepts in the News. Identify the major un-resolved issue discussed in today& news. List at th...
Using and Understanding Mathematics: A Quantitative Reasoning Approach (6th Edition)
In Exercises 5-36, express all probabilities as fractions.
23. Combination Lock The typical combination lock us...
Elementary Statistics
1. combination of numbers, variables, and operation symbols is called an algebraic______.
Algebra and Trigonometry (6th Edition)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.Similar questions
- (a) State, without proof, Cauchy's theorem, Cauchy's integral formula and Cauchy's integral formula for derivatives. Your answer should include all the conditions required for the results to hold. (8 marks) (b) Let U{z EC: |z| -1}. Let 12 be the triangular contour with vertices at 0, 2-2 and 2+2i, parametrized in the anticlockwise direction. Calculate dz. You must check the conditions of any results you use. (d) Let U C. Calculate Liz-1ym dz, (z - 1) 10 (5 marks) where 2 is the same as the previous part. You must check the conditions of any results you use. (4 marks)arrow_forward(a) Suppose a function f: C→C has an isolated singularity at wЄ C. State what it means for this singularity to be a pole of order k. (2 marks) (b) Let f have a pole of order k at wЄ C. Prove that the residue of f at w is given by 1 res (f, w): = Z dk (k-1)! >wdzk−1 lim - [(z — w)* f(z)] . (5 marks) (c) Using the previous part, find the singularity of the function 9(z) = COS(πZ) e² (z - 1)²' classify it and calculate its residue. (5 marks) (d) Let g(x)=sin(211). Find the residue of g at z = 1. (3 marks) (e) Classify the singularity of cot(z) h(z) = Z at the origin. (5 marks)arrow_forward1. Let z = x+iy with x, y Є R. Let f(z) = u(x, y) + iv(x, y) where u(x, y), v(x, y): R² → R. (a) Suppose that f is complex differentiable. State the Cauchy-Riemann equations satisfied by the functions u(x, y) and v(x,y). (b) State what it means for the function (2 mark) u(x, y): R² → R to be a harmonic function. (3 marks) (c) Show that the function u(x, y) = 3x²y - y³ +2 is harmonic. (d) Find a harmonic conjugate of u(x, y). (6 marks) (9 marks)arrow_forward
- Please could you provide a step by step solutions to this question and explain every step.arrow_forwardCould you please help me with question 2bii. If possible could you explain how you found the bounds of the integral by using a graph of the region of integration. Thanksarrow_forwardLet A be a vector space with basis 1, a, b. Which (if any) of the following rules turn A into an algebra? (You may assume that 1 is a unit.) (i) a² = a, b² = ab = ba = 0. (ii) a²=b, b² = ab = ba = 0. (iii) a²=b, b² = b, ab = ba = 0.arrow_forward
- No chatgpt pls will upvotearrow_forward= 1. Show (a) Let G = Z/nZ be a cyclic group, so G = {1, 9, 92,...,g" } with g": that the group algebra KG has a presentation KG = K(X)/(X” — 1). (b) Let A = K[X] be the algebra of polynomials in X. Let V be the A-module with vector space K2 and where the action of X is given by the matrix Compute End(V) in the cases (i) x = p, (ii) xμl. (67) · (c) If M and N are submodules of a module L, prove that there is an isomorphism M/MON (M+N)/N. (The Second Isomorphism Theorem for modules.) You may assume that MON is a submodule of M, M + N is a submodule of L and the First Isomorphism Theorem for modules.arrow_forward(a) Define the notion of an ideal I in an algebra A. Define the product on the quotient algebra A/I, and show that it is well-defined. (b) If I is an ideal in A and S is a subalgebra of A, show that S + I is a subalgebra of A and that SnI is an ideal in S. (c) Let A be the subset of M3 (K) given by matrices of the form a b 0 a 0 00 d Show that A is a subalgebra of M3(K). Ꮖ Compute the ideal I of A generated by the element and show that A/I K as algebras, where 0 1 0 x = 0 0 0 001arrow_forward
- (a) Let HI be the algebra of quaternions. Write out the multiplication table for 1, i, j, k. Define the notion of a pure quaternion, and the absolute value of a quaternion. Show that if p is a pure quaternion, then p² = -|p|². (b) Define the notion of an (associative) algebra. (c) Let A be a vector space with basis 1, a, b. Which (if any) of the following rules turn A into an algebra? (You may assume that 1 is a unit.) (i) a² = a, b²=ab = ba 0. (ii) a² (iii) a² = b, b² = abba = 0. = b, b² = b, ab = ba = 0. (d) Let u1, 2 and 3 be in the Temperley-Lieb algebra TL4(8). ገ 12 13 Compute (u3+ Augu2)² where A EK and hence find a non-zero x € TL4 (8) such that ² = 0.arrow_forwardQ1: Solve the system x + x = t², x(0) = (9)arrow_forwardCo Given show that Solution Take home Су-15 1994 +19 09/2 4 =a log суто - 1092 ж = a-1 2+1+8 AI | SHOT ON S4 INFINIX CAMERAarrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageAlgebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning
- Glencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw HillCollege Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage LearningAlgebra: Structure And Method, Book 1AlgebraISBN:9780395977224Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. ColePublisher:McDougal Littell
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:Cengage Learning
Glencoe Algebra 1, Student Edition, 9780079039897...
Algebra
ISBN:9780079039897
Author:Carter
Publisher:McGraw Hill
College Algebra (MindTap Course List)
Algebra
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:Cengage Learning
Algebra: Structure And Method, Book 1
Algebra
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:McDougal Littell
Implicit Differentiation Explained - Product Rule, Quotient & Chain Rule - Calculus; Author: The Organic Chemistry Tutor;https://www.youtube.com/watch?v=LGY-DjFsALc;License: Standard YouTube License, CC-BY