Mathematical Methods in the Physical Sciences
3rd Edition
ISBN: 9780471198260
Author: Mary L. Boas
Publisher: Wiley, John & Sons, Incorporated
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Question
Chapter 3.7, Problem 4P
To determine
If
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Students have asked these similar questions
Can you answer this question and give step by step and why and how to get it. Can you write it (numerical method)
Construct tables showing the values of alI the Dirichlet characters mod k fork = 8,9, and 10.
(please show me result in a table and the equation in mathematical format.)
Example: For what odd primes p is 11 a quadratic residue modulo p?
Solution:
This is really asking "when is (11 | p) =1?"
First, 11 = 3 (mod 4). To use LQR, consider two cases p = 1 or 3 (mod 4):
p=1 We have 1 = (11 | p) = (p | 11), so p is a quadratic residue modulo 11. By
brute force:
121, 224, 3² = 9, 4² = 5, 5² = 3 (mod 11)
so the quadratic residues mod 11 are 1,3,4,5,9.
Using CRT for p = 1 (mod 4) & p = 1,3,4,5,9 (mod 11).
p = 1
(mod 4)
&
p = 1
(mod 11
gives p
1
(mod 44).
p = 1
(mod 4)
&
p = 3
(mod 11)
gives p25
(mod 44).
p = 1
(mod 4)
&
p = 4
(mod 11)
gives p=37
(mod 44).
p = 1
(mod 4)
&
p = 5
(mod 11)
gives p
5
(mod 44).
p = 1
(mod 4)
&
p=9
(mod 11)
gives p
9
(mod 44).
So p =1,5,9,25,37 (mod 44).
Chapter 3 Solutions
Mathematical Methods in the Physical Sciences
Ch. 3.2 - The first equation in (2.6) written out in detail...Ch. 3.2 - Prob. 2PCh. 3.2 - For each of the following problems write and row...Ch. 3.2 - For each of the following problems write and row...Ch. 3.2 - For each of the following problems write and row...Ch. 3.2 - For each of the following problems write and row...Ch. 3.2 - For each of the following problems write and row...Ch. 3.2 - For each of the following problems write and row...Ch. 3.2 - For each of the following problems write and row...Ch. 3.2 - For each of the following problems write and row...
Ch. 3.2 - For each of the following problems write and row...Ch. 3.2 - For each of the following problems write and row...Ch. 3.2 - For each of the following problems write and row...Ch. 3.2 - For each of the following problems write and row...Ch. 3.2 - Find the rank of each of the following matrices....Ch. 3.2 - Find the rank of each of the following matrices....Ch. 3.2 - Find the rank of each of the following matrices....Ch. 3.2 - Find the rank of each of the following matrices....Ch. 3.3 - Evaluate the determinants in Problems 1 to 6 by...Ch. 3.3 - Evaluate the determinants in Problems 1 to 6 by...Ch. 3.3 - Evaluate the determinants in Problems 1 to 6 by...Ch. 3.3 - Evaluate the determinants in Problems 1 to 6 by...Ch. 3.3 - Evaluate the determinants in Problems 1 to 6 by...Ch. 3.3 - Evaluate the determinants in Problems 1 to 6 by...Ch. 3.3 - Prove the following by appropriate manipulations...Ch. 3.3 - Prob. 8PCh. 3.3 - Show without computation that the following...Ch. 3.3 - A determinant or a square matrix is called...Ch. 3.3 - In Problems 11 and 12 evaluate the determminants....Ch. 3.3 - In Problems 11 and 12 evaluate the determminants....Ch. 3.3 - Show that cos1012cos1012cos=cos3Ch. 3.3 - Show that the n-rowed determinant Hint: Expand...Ch. 3.3 - Use Cramers rule to solve Problem 2.3 and 2.11.Ch. 3.3 - In the following set of equations (from a quantum...Ch. 3.3 - Use Cramers rule to solve for x and t the Lorentz...Ch. 3.3 - Find z by Cramers rule:...Ch. 3.4 - Draw diagrams and prove (4.1).Ch. 3.4 - Given the vectors making the given angles With...Ch. 3.4 - Use vectors to prove the following theorems from...Ch. 3.4 - Use vectors to prove the following theorems from...Ch. 3.4 - Use vectors to prove the following theorems from...Ch. 3.4 - Use vectors to prove the following theorems from...Ch. 3.4 - Use vectors to prove the following theorems from...Ch. 3.4 - Use vectors to prove the following theorems from...Ch. 3.4 - Let A=2i+3j and B=4i4j. Show graphically, and find...Ch. 3.4 - If A+B =4j-i and A —B=i+3j, find A and B...Ch. 3.4 - Let 3i—j+4k, 7j—2k, i—3j+k be three vectors...Ch. 3.4 - Find the angle between the vectors A=2i+j2k and...Ch. 3.4 - If A = 4i-3k and B = —2i+2j— k, find the...Ch. 3.4 - Prob. 14PCh. 3.4 - Let A = 2i—j+2k. (a) Find a unit vector in the...Ch. 3.4 - Prob. 16PCh. 3.4 - Find three vectors (none of them parallel to a...Ch. 3.4 - Prob. 18PCh. 3.4 - Prob. 19PCh. 3.4 - Fine a vector perpendicular to both i+j and i-2k.Ch. 3.4 - Show that B|A|+A|B| and A|B|-B|A| are orthogonal.Ch. 3.4 - Square (A + B); interpret your result...Ch. 3.4 - If A = 2i—3j+ k and A • B = 0, does it follow...Ch. 3.4 - What is the value of (AB)2+(AB)2 ? Comment: This...Ch. 3.4 - Use vectors as in Problems 3 to 8, and also the...Ch. 3.4 - Use vectors as in Problems 3 to 8, and also the...Ch. 3.4 - Use vectors as in Problems 3 to 8, and also the...Ch. 3.4 - Use vectors as in Problems 3 to 8, and also the...Ch. 3.5 - In Problems 1 to 5, all lines are in the (x,y)...Ch. 3.5 - In Problems 1 to 5, all lines are in the (x,y)...Ch. 3.5 - In Problems 1 to 5, all lines are in the (x,y)...Ch. 3.5 - In Problems 1 to 5, all lines are in the (x,y)...Ch. 3.5 - In Problems 1 to 5, all lines are in the (x,y)...Ch. 3.5 - Prob. 6PCh. 3.5 - Prob. 7PCh. 3.5 - Prob. 8PCh. 3.5 - Prob. 9PCh. 3.5 - Prob. 10PCh. 3.5 - Prob. 11PCh. 3.5 - Prob. 12PCh. 3.5 - Find the symmetric equations (5.6) or (5.7) and...Ch. 3.5 - Prob. 14PCh. 3.5 - Prob. 15PCh. 3.5 - Prob. 16PCh. 3.5 - Prob. 17PCh. 3.5 - Prob. 18PCh. 3.5 - Prob. 19PCh. 3.5 - Find the symmetric equations (5.6) or (5.7) and...Ch. 3.5 - In Problems 21 to 23, find the angle between the...Ch. 3.5 - In Problems 21 to 23, find the angle between the...Ch. 3.5 - In Problems 21 to 23, find the angle between the...Ch. 3.5 - Find a point on both the planes (that is, on their...Ch. 3.5 - As in Problem 24, find the equations of the line...Ch. 3.5 - Prob. 26PCh. 3.5 - Find the equation of the plane through (2, 3,...Ch. 3.5 - Find the equation of the plane through (-4, -1, 2)...Ch. 3.5 - Find a point on the plane 2x — y — z = 13....Ch. 3.5 - Find the distance from the origin to the plane 3x...Ch. 3.5 - Find the distance from (-2, 4, 5) to the plane...Ch. 3.5 - Find the distance from (3, -1, 2) to the plane 5x...Ch. 3.5 - Findthe perpendicular distance between the two...Ch. 3.5 - Find the distance (perpendicular is understood)...Ch. 3.5 - Find the distance (2,5,1) to the line in Problem...Ch. 3.5 - Find the distance (3,2,5) to the line in Problem...Ch. 3.5 - Determine whether the lines x12=y+31=z43 and...Ch. 3.5 - Find the angle between the lines in Problem 37.Ch. 3.5 - In Problems 39 and 40, show that the given lines...Ch. 3.5 - In Problems 39 and 40, show that the given lines...Ch. 3.5 - In Problems 41 to 44, find the distance between...Ch. 3.5 - In Problems 41 to 44, find the distance between...Ch. 3.5 - In Problems 41 to 44, find the distance between...Ch. 3.5 - In Problems 41 to 44, find the distance between...Ch. 3.5 - A particle is traveling along the line (x — 3)/2...Ch. 3.6 - In Problems 1 to 3, find AB,BA,A+B,AB,A2,B2,5A,3B....Ch. 3.6 - In Problems 1 to 3, find AB,BA,A+B,AB,A2,B2,5A,3B....Ch. 3.6 - In Problems 1 to 3, find AB,BA,A+B,AB,A2,B2,5A,3B....Ch. 3.6 - Given the matrices A=23142105, B=241131,...Ch. 3.6 - Compute the product of each of the matrices in...Ch. 3.6 - The Pauli spin in quantum mechanics are...Ch. 3.6 - Find the matrix product 23142112 By evaluating...Ch. 3.6 - Show, by multiplying the matrices, that the...Ch. 3.6 - Find AB and BA given A=1236,B=10452. Observe that...Ch. 3.6 - Prob. 10PCh. 3.6 - Show that the unit matrix I has the property that...Ch. 3.6 - For the matrices in Example 3, verify that MM—1...Ch. 3.6 - In Problems 13 to 16, use (6.13) to find the...Ch. 3.6 - In Problems 13 to 16, use (6.13) to find the...Ch. 3.6 - In Problems 13 to 16, use (6.13) to find the...Ch. 3.6 - In Problems 13 to 16, use (6.13) to find the...Ch. 3.6 - Given the matrices A=111401420,B=101211212 (a)...Ch. 3.6 - Problem 17(b) is a special case of the general...Ch. 3.6 - In Problems 19 to 22, solve each set of equations...Ch. 3.6 - In Problems 19 to 22, solve each set of equations...Ch. 3.6 - In Problems 19 to 22, solve each set of equations...Ch. 3.6 - In Problems 19 to 22, solve each set of equations...Ch. 3.6 - Verify formula (6.13). Hint: Consider the product...Ch. 3.6 - Use the method of solving simultaneous equations...Ch. 3.6 - Verify (6.14) by multiplying the matrices and...Ch. 3.6 - In (6.14), let ==/2 and verify the result...Ch. 3.6 - Do Problem 26 if =/2,=/4.Ch. 3.6 - Verify the calculations in (6.15), (6.16), and...Ch. 3.6 - Show that if A and B are matrices which dont...Ch. 3.6 - For the Pauli spin matrix A in Problem 6, find the...Ch. 3.6 - Repeat Problem 30 for the Pauli spin matrix C in...Ch. 3.6 - For the Pauli spin matrix B in Problem 6, find eiB...Ch. 3.7 - Prob. 1PCh. 3.7 - Are the following linear functions? Prove your...Ch. 3.7 - Are the following linear functions? Prove your...Ch. 3.7 - Prob. 4PCh. 3.7 - Are the following linear vector functions? Prove...Ch. 3.7 - Are the following linear vector functions? Prove...Ch. 3.7 - Are the following operators linear? Definite...Ch. 3.7 - Are the following operators linear? Find the...Ch. 3.7 - Are the following operators linear? Find the...Ch. 3.7 - Are the following operators linear? Find the...Ch. 3.7 - Are the following operators linear? Find the...Ch. 3.7 - Are the following operators linear? Let D stand...Ch. 3.7 - Are the following operators linear? (a) As in...Ch. 3.7 - Are the following operators linear? Find the...Ch. 3.7 - Are the following operators linear? Find the...Ch. 3.7 - Are the following operators linear? Find the...Ch. 3.7 - Are the following operators linear? Find the...Ch. 3.7 - With the cross product of two vectors defined by...Ch. 3.7 - If multiply a complex number z=ri by ei, we get...Ch. 3.7 - Verify equations (7.13) using Figure 7.5. Hints:...Ch. 3.7 - Do the details Of Example 3 as follows: Verify...Ch. 3.7 - Let each of the following matrices represent an...Ch. 3.7 - Let each of the following matrices represent an...Ch. 3.7 - Let each of the following matrices represent an...Ch. 3.7 - Let each of the following matrices represent an...Ch. 3.7 - Let each of the following matrices represent an...Ch. 3.7 - Let each of the following matrices represent an...Ch. 3.7 - Write the matrices which produce a rotation about...Ch. 3.7 - Construct the matrix corresponding to a rotation...Ch. 3.7 - For the matrices G and K in (7.21), find the...Ch. 3.7 - To see a physical example of non-commuting...Ch. 3.7 - For each of the following matrices, find its...Ch. 3.7 - For each of the following matrices, find its...Ch. 3.7 - For each of the following matrices, find its...Ch. 3.7 - For each of the following matrices, find its...Ch. 3.8 - Write each of the vectors (8.1) as a linear...Ch. 3.8 - In Problems 2 to 4, find out whether the given...Ch. 3.8 - In Problems 2 to 4, find out whether the given...Ch. 3.8 - In Problems 2 to 4, find out whether the given...Ch. 3.8 - Show that any vector V in a plane can be written...Ch. 3.8 - Use Problem 5 to write V = 3i + 5j as a linear...Ch. 3.8 - As in Problem 6, write V = 4i-5j in terms of the...Ch. 3.8 - In Problems 8 to 15, use (8.5) to show that the...Ch. 3.8 - In Problems 8 to 15, use (8.5) to show that the...Ch. 3.8 - In Problems 8 to 15, use (8.5) to show that the...Ch. 3.8 - In Problems 8 to 15, use (8.5) to show that the...Ch. 3.8 - In Problems 8 to 15, use (8.5) to show that the...Ch. 3.8 - In Problems 8 to 15, use (8.5) to show that the...Ch. 3.8 - In Problems 8 to 15, use (8.5) to show that the...Ch. 3.8 - In Problems 8 to 15, use (8.5) to show that the...Ch. 3.8 - Prove that if the Wronskian (8.5) is not...Ch. 3.8 - In Problems 17 to 20, solve the sets of...Ch. 3.8 - In Problems 17 to 20, solve the sets of...Ch. 3.8 - In Problems 17 to 20, solve the sets of...Ch. 3.8 - In Problems 17 to 20, solve the sets of...Ch. 3.8 - Find a condition for four points in space to lie...Ch. 3.8 - Find a condition for three lines in a plane to...Ch. 3.8 - Using (8.9), find the values of such that the...Ch. 3.8 - Using (8.9), find the values of such that the...Ch. 3.8 - Using (8.9), find the values of such that the...Ch. 3.8 - For each of the following, write the solution in...Ch. 3.8 - For each of the following, write the solution in...Ch. 3.8 - For each of the following, write the solution in...Ch. 3.9 - Use index notation as in 9.9 to prove the second...Ch. 3.9 - Use index notation to prove the distributive law...Ch. 3.9 - Given the following matrix, find the transpose,...Ch. 3.9 - Repeat Problem 3 given A=02i1i20300.Ch. 3.9 - Show that the product AAT is a symmetric matrix.Ch. 3.9 - Give numerical examples of: a symmetric matrix; a...Ch. 3.9 - Write each of the items in the second column of...Ch. 3.9 - Prove that ABt=BtAt. Hint: see 9.10. Verify 9.11,...Ch. 3.9 - In 9.1 we have defined the adjoint of a matrix as...Ch. 3.9 - Show that if a matrix is orthogonal and its...Ch. 3.9 - Show that a real Hermitian matrix is symmetric....Ch. 3.9 - Show that the definition of a Hermitian matrix...Ch. 3.9 - Show that the following matrix is a unitary...Ch. 3.9 - Prob. 14PCh. 3.9 - Show that the Pauli spin matrices (Problem 6.6)...Ch. 3.9 - Let Cij=1i+jMij be the cofactor of element aij in...Ch. 3.9 - Show that if A and B are symmetric, then AB is not...Ch. 3.9 - If A and B are symmetric matrices, show that their...Ch. 3.9 - Prove that TrAB=TrBA. Hint: see proof of (9.13)....Ch. 3.9 - Show that the determinant of a unitary matrix is a...Ch. 3.9 - Show that the transpose of a sum of matrices is...Ch. 3.9 - Show that a unitary matrix is a normal matrix,...Ch. 3.9 - Show that the following matrices are Hermitian...Ch. 3.9 - Show that an orthogonal transformation preserves...Ch. 3.9 - Show that the inverse of an orthogonal matrix is...Ch. 3.10 - Find the distance between the points 4,1,2,7 and...Ch. 3.10 - For the given sets of vectors, find the dimension...Ch. 3.10 - (a) Find the cosines of the angles between pairs...Ch. 3.10 - For each given set of basis vectors, use the...Ch. 3.10 - By 10.6 and 10.7, find the norms of A and B and...Ch. 3.10 - Write out the proof of the Schwarz inequality 10.9...Ch. 3.10 - Show that, in n-dimensional space, any n+1 vectors...Ch. 3.10 - Show that two different sets of basis vectors for...Ch. 3.10 - Write equations 10.6 to 10.9 in matrix form as...Ch. 3.10 - Prove that A+BA+B. This is called the triangle...Ch. 3.11 - Verify 11.7. Also verify 11.12 and find the...Ch. 3.11 - Verify that the two eigenvectors in 11.8 are...Ch. 3.11 - If C is orthogonal and M is symmetric, show that...Ch. 3.11 - Find the inverse of the rotation matrix in 7.13;...Ch. 3.11 - Show that the C matrix in 11.10 does represent a...Ch. 3.11 - Show that if C is a matrix whose columns are the...Ch. 3.11 - Generalize Problem 6 to three dimensions; to n...Ch. 3.11 - Show that under the transformation 11.1, all...Ch. 3.11 - Show that detC1MC=detM. Hints: See 6.6. What is...Ch. 3.11 - Show that TrC1MC=TrM. Hint: see (9.13). Thus show...Ch. 3.11 - Find the inverse of the transformation...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Find the eigenvalues and eigenvectors of the...Ch. 3.11 - Let each of the following matrices M describe a...Ch. 3.11 - Let each of the following matrices M describe a...Ch. 3.11 - Let each of the following matrices M describe a...Ch. 3.11 - Let each of the following matrices M describe a...Ch. 3.11 - Let each of the following matrices M describe a...Ch. 3.11 - Let each of the following matrices M describe a...Ch. 3.11 - Find the eigenvalues and eigenvectors of the real...Ch. 3.11 - By multiplying out M=CDC1 the diagonal matrix...Ch. 3.11 - The characteristic equation for a second-order...Ch. 3.11 - Verify the eigenvalues and eigenvectors of matrix...Ch. 3.11 - Starting with 11.23, obtain 11.24. Hints: Take the...Ch. 3.11 - Verify equation 11.25. Hint: Remember from Section...Ch. 3.11 - Write out the detailed proof of 11.27. Hint:...Ch. 3.11 - Verify the details as indicated in diagonalizing H...Ch. 3.11 - Verify that each of the following matrices is...Ch. 3.11 - Verify that each of the following matrices is...Ch. 3.11 - Verify that each of the following matrices is...Ch. 3.11 - Verify that each of the following matrices is...Ch. 3.11 - Verify the details in the discussion of the...Ch. 3.11 - We have seen that an orthogonal matrix with...Ch. 3.11 - Find a unitary matrix U which diagonalizes A in...Ch. 3.11 - Show that an orthogonal matrix M with all real...Ch. 3.11 - Verify the results for F in the discussion of...Ch. 3.11 - Show that the trace of a rotation matrix equals...Ch. 3.11 - Show that each of the following matrices is...Ch. 3.11 - Show that each of the following matrices is...Ch. 3.11 - Show that each of the following matrices is...Ch. 3.11 - Show that each of the following matrices is...Ch. 3.11 - Show that each of the following matrices is...Ch. 3.11 - Show that each of the following matrices is...Ch. 3.11 - Show that if D is a diagonal matrix, then Dn is...Ch. 3.11 - Note in Section 6 [see (6.15)] that, for the given...Ch. 3.11 - Repeat the last part of Problem 58 for the matrix...Ch. 3.11 - The Caley-Hamilton theorem states that A matrix...Ch. 3.11 - At the end of Section 9 we proved that if H is a...Ch. 3.11 - Show that if matrices F and G can be diagonalized...Ch. 3.12 - Verify that 12.2 multiplied out is 12.1.Ch. 3.12 - Find the equations of the following conics and...Ch. 3.12 - Find the equations of the following conics and...Ch. 3.12 - Find the equations of the following conics and...Ch. 3.12 - Find the equations of the following conics and...Ch. 3.12 - Find the equations of the following conics and...Ch. 3.12 - Find the equations of the following conics and...Ch. 3.12 - Carry through the details of Example 2 to find the...Ch. 3.12 - For Problems 2 to 7, find the rotation matrix C...Ch. 3.12 - Verify equations 12.13 and 12.14. Solve 12.15 to...Ch. 3.12 - Verify the details of Example 4, equations 12.18...Ch. 3.12 - Verify the details of Example 5, equations 12.26...Ch. 3.12 - Verify the details of Example 6, equations 12.37...Ch. 3.12 - Find the characteristic frequencies and the...Ch. 3.12 - Find the characteristic frequencies and the...Ch. 3.12 - Find the characteristic frequencies and the...Ch. 3.12 - Find the characteristic frequencies and the...Ch. 3.12 - Find the characteristic frequencies and the...Ch. 3.12 - Find the characteristic frequencies and the...Ch. 3.12 - Carry through the details of Example 7.Ch. 3.12 - Find the characteristic frequencies and the...Ch. 3.12 - Find the characteristic frequencies and the...Ch. 3.12 - Find the characteristic frequencies and the...Ch. 3.13 - Write the four rotation matrices for rotations of...Ch. 3.13 - Following the text discussion of the cyclic group...Ch. 3.13 - Prob. 3PCh. 3.13 - Show that the matrices...Ch. 3.13 - Consider the group of order 4 with unit element I...Ch. 3.13 - Consider the integers 0, 1, 2, 3 under addition...Ch. 3.13 - Consider the set of numbers 1, 3, 5, 7 with...Ch. 3.13 - Verify 13.3 and 13.4. Hints: For the rotation and...Ch. 3.13 - Show that any cyclic group is Abelian. Hint: Does...Ch. 3.13 - Prob. 10PCh. 3.13 - Do Problem 10 for a rectangle. Note that now only...Ch. 3.13 - Verify 13.5 and then also show that A, B are the...Ch. 3.13 - Using the discussion of simultaneous...Ch. 3.13 - Use the multiplication table you found in Problem...Ch. 3.13 - By Problem 13, you know that the matrices in...Ch. 3.13 - Do Problem 15 for the group of matrices you found...Ch. 3.13 - Verify that the sets listed in 13.7c are groups.Ch. 3.13 - Prob. 18PCh. 3.13 - Verify that the sets listed in 13.7e are groups....Ch. 3.13 - Is the set of all orthogonal 3-by-3 matrices with...Ch. 3.13 - Prob. 21PCh. 3.14 - Verify the statements indicated in Examples 1 to 5...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.14 - For each of the following sets, either verify (as...Ch. 3.15 - Show that if each element of one row (or column)...Ch. 3.15 - What is wrong with the following argument? If we...Ch. 3.15 - Find the equations of the line through the points...Ch. 3.15 - Given the line r=3ij+2i+j2kt: Find the equation of...Ch. 3.15 - Write the equations of a straight line through the...Ch. 3.15 - Derive the formula D=ax0+by0+cz0da2+b2+c2 for the...Ch. 3.15 - Given the matrices A, B, C below, find or mark as...Ch. 3.15 - Given A=102ii3010i, find AT,A,At,A1.Ch. 3.15 - The following matrix product is used in discussing...Ch. 3.15 - The following matrix product is used in discussing...Ch. 3.15 - There is a one-to-one correspondence between...Ch. 3.15 - The vectors A=aibj and B=ci+dj form two sides of a...Ch. 3.15 - The plane 2x+3y+6z=6 intersects the coordinate...Ch. 3.15 - In Problems 14 to 17, multiply matrices to find...Ch. 3.15 - In Problems 14 to 17, multiply matrices to find...Ch. 3.15 - In Problems 14 to 17, multiply matrices to find...Ch. 3.15 - In Problems 14 to 17, multiply matrices to find...Ch. 3.15 - Prob. 18MPCh. 3.15 - Find the eigenvalues and eigenvectors of the...Ch. 3.15 - Find the eigenvalues and eigenvectors of the...Ch. 3.15 - Find the eigenvalues and eigenvectors of the...Ch. 3.15 - Find the eigenvalues and eigenvectors of the...Ch. 3.15 - Find the eigenvalues and eigenvectors of the...Ch. 3.15 - Find the eigenvalues and eigenvectors of the...Ch. 3.15 - Find the C matrix which diagonalizes the matrix M...Ch. 3.15 - Repeat Problem 25 for Problem 19. Find the C...Ch. 3.15 - In Problems 27 to 30, rotate the given quadric...Ch. 3.15 - In Problems 27 to 30, rotate the given quadric...Ch. 3.15 - In Problems 27 to 30, rotate the given quadric...Ch. 3.15 - In Problems 27 to 30, rotate the given quadric...Ch. 3.15 - Find the characteristic vibration frequencies of a...Ch. 3.15 - Do Problem 31 if the spring constants are...Ch. 3.15 - Prove the Caley-Hamilton theorem (Problem 11.60)...Ch. 3.15 - In problems 6.30 and 6.31, you found the matrices...Ch. 3.15 - Show that a square matrix A has an inverse if and...Ch. 3.15 - Write the three 3 by 3 matrices for 180 rotations...Ch. 3.15 - Show that for a given irreducible representation...Ch. 3.15 - For a cyclic group, show that every element is a...
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- Kyoko has $10,000 that she wants to invest. Her bank has several accounts to choose from. Her goal is to have $15,000 by the time she finishes graduate school in 7 years. To the nearest hundredth of a percent, what should her minimum annual interest rate be in order to reach her goal assuming they compound daily? (Hint: solve the compound interest formula for the intrerest rate. Also, assume there are 365 days in a year) %arrow_forwardTest the claim that a student's pulse rate is different when taking a quiz than attending a regular class. The mean pulse rate difference is 2.7 with 10 students. Use a significance level of 0.005. Pulse rate difference(Quiz - Lecture) 2 -1 5 -8 1 20 15 -4 9 -12arrow_forwardThere are three options for investing $1150. The first earns 10% compounded annually, the second earns 10% compounded quarterly, and the third earns 10% compounded continuously. Find equations that model each investment growth and use a graphing utility to graph each model in the same viewing window over a 20-year period. Use the graph to determine which investment yields the highest return after 20 years. What are the differences in earnings among the three investment? STEP 1: The formula for compound interest is A = nt = P(1 + − − ) n², where n is the number of compoundings per year, t is the number of years, r is the interest rate, P is the principal, and A is the amount (balance) after t years. For continuous compounding, the formula reduces to A = Pert Find r and n for each model, and use these values to write A in terms of t for each case. Annual Model r=0.10 A = Y(t) = 1150 (1.10)* n = 1 Quarterly Model r = 0.10 n = 4 A = Q(t) = 1150(1.025) 4t Continuous Model r=0.10 A = C(t) =…arrow_forward
- The following ordered data list shows the data speeds for cell phones used by a telephone company at an airport: A. Calculate the Measures of Central Tendency from the ungrouped data list. B. Group the data in an appropriate frequency table. C. Calculate the Measures of Central Tendency using the table in point B. D. Are there differences in the measurements obtained in A and C? Why (give at least one justified reason)? I leave the answers to A and B to resolve the remaining two. 0.8 1.4 1.8 1.9 3.2 3.6 4.5 4.5 4.6 6.2 6.5 7.7 7.9 9.9 10.2 10.3 10.9 11.1 11.1 11.6 11.8 12.0 13.1 13.5 13.7 14.1 14.2 14.7 15.0 15.1 15.5 15.8 16.0 17.5 18.2 20.2 21.1 21.5 22.2 22.4 23.1 24.5 25.7 28.5 34.6 38.5 43.0 55.6 71.3 77.8 A. Measures of Central Tendency We are to calculate: Mean, Median, Mode The data (already ordered) is: 0.8, 1.4, 1.8, 1.9, 3.2, 3.6, 4.5, 4.5, 4.6, 6.2, 6.5, 7.7, 7.9, 9.9, 10.2, 10.3, 10.9, 11.1, 11.1, 11.6, 11.8, 12.0, 13.1, 13.5, 13.7, 14.1, 14.2, 14.7, 15.0, 15.1, 15.5,…arrow_forwardA tournament is a complete directed graph, for each pair of vertices x, y either (x, y) is an arc or (y, x) is an arc. One can think of this as a round robin tournament, where the vertices represent teams, each pair plays exactly once, with the direction of the arc indicating which team wins. (a) Prove that every tournament has a direct Hamiltonian path. That is a labeling of the teams V1, V2,..., Un so that vi beats Vi+1. That is a labeling so that team 1 beats team 2, team 2 beats team 3, etc. (b) A digraph is strongly connected if there is a directed path from any vertex to any other vertex. Equivalently, there is no partition of the teams into groups A, B so that every team in A beats every team in B. Prove that every strongly connected tournament has a directed Hamiltonian cycle. Use this to show that for any team there is an ordering as in part (a) for which the given team is first. (c) A king in a tournament is a vertex such that there is a direct path of length at most 2 to any…arrow_forwardUse a graphing utility to find the point of intersection, if any, of the graphs of the functions. Round your result to three decimal places. (Enter NONE in any unused answer blanks.) y = 100e0.01x (x, y) = y = 11,250 ×arrow_forward
- how to construct the following same table?arrow_forwardThe following is known. The complete graph K2t on an even number of vertices has a 1- factorization (equivalently, its edges can be colored with 2t - 1 colors so that the edges incident to each vertex are distinct). This implies that the complete graph K2t+1 on an odd number of vertices has a factorization into copies of tK2 + K₁ (a matching plus an isolated vertex). A group of 10 people wants to set up a 45 week tennis schedule playing doubles, each week, the players will form 5 pairs. One of the pairs will not play, the other 4 pairs will each play one doubles match, two of the pairs playing each other and the other two pairs playing each other. Set up a schedule with the following constraints: Each pair of players is a doubles team exactly 4 times; during those 4 matches they see each other player exactly once; no two doubles teams play each other more than once. (a) Find a schedule. Hint - think about breaking the 45 weeks into 9 blocks of 5 weeks. Use factorizations of complete…arrow_forward. The two person game of slither is played on a graph. Players 1 and 2 take turns, building a path in the graph. To start, Player 1 picks a vertex. Player 2 then picks an edge incident to the vertex. Then, starting with Player 1, players alternate turns, picking a vertex not already selected that is adjacent to one of the ends of the path created so far. The first player who cannot select a vertex loses. (This happens when all neighbors of the end vertices of the path are on the path.) Prove that Player 2 has a winning strategy if the graph has a perfect matching and Player 1 has a winning strategy if the graph does not have a perfect matching. In each case describe a strategy for the winning player that guarantees that they will always be able to select a vertex. The strategy will be based on using a maximum matching to decide the next choice, and will, for one of the cases involve using the fact that maximality means no augmenting paths. Warning, the game slither is often described…arrow_forward
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