Mathematical Methods in the Physical Sciences
3rd Edition
ISBN: 9780471198260
Author: Mary L. Boas
Publisher: Wiley, John & Sons, Incorporated
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Textbook Question
Chapter 3.7, Problem 32P
For each of the following matrices, find its determinant to see whether it produces a rotation or a reflection. If a rotation, find the axis and angle of rotation. If a reflection, find the reflecting plane and the rotation (if any) about the normal to this plane.
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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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- (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
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