Physics 3513, Homework #4 (due 9/19)
The numbers in parentheses after the problem number indicate points for each problem.
P1(10) (Boas 5.10) Find symmetric and parametric equations of a line through (3,4,−1) and
parallel to 2i−3j+ 6k.
P2(10) (Boas 5.35) Find the distance from (2,5,1) to the line in the previous problem.
P3(10) (Boas 7.22) Show that matrix 1
√2 1 1
−1 1 !is orthogonal, find its determinant, and
find the rotation angle or the line of reflection.
P4(10) (Boas 8.13) Calculate the Wronskian of functions f1(x) = sin x,f2(x) = sin 2xto
show that they are linearly independent.
P5(10) (Boas 8.17) Solve
x−2y+ 3z= 0
x+ 4y−6z= 0
2x+ 2y−3z= 0
P6(10) (Boas 11.13) Find the eigenvalues and eigenvectors of 2 2
2−1!
P7(10) (Boas, from 11.42) Verify that matrix 3 1 −i
1 + i2!is Hermitian and find its
eigenvalues and eigenvectors.
P8(10) Find the eigenvalues and eigenvectors of
(a) 1
√2 1ε
ε1!, (b) 1
√2 1 1
ε21!.
In both cases, find the angle between the two eigenvectors as ε→0.
P9(20) (a) Show that for any given matrix M the sum of all eigenvalues is equal to Tr M
(Problem Boas 11.10), and the product of all eigenvalues is equal to det M
(Problem Boas 11.9).
(b) Using the results of (a), find the sum and the product of all eigenvalues for the
matrix
A =
0 0 0 0 1
0 1 0 0 0
0 0 0 1 0
1 0 0 0 0
0 0 1 0 0
(c) Find the eigenvalues of A and check your results of (b).
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