Math 121: Linear Algebra and Applications Solution Set 1
Posted: Fri. Oct. 12, 2007
Prof. Lydia Bieri
Written by: Luca Candelori
Exercise 1. (a) TRUE. This is one of the vector space axioms (VS 3). (b) FALSE. We proved this in class but here is a short pr
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and multiplying these two series term by term. To be precise, we may multiply each
term in the first series by 1, then each term in that series by -2, etc. The following
systematic approach is suggested, where like powers of z are assembled vertically so
EXAMPLE 2. The origin is a singular point of the principal branch (Sec. 30)
Logz=lnr+iO
(r>O,-n<O<x)
of the logarithmic function. It is not, however, an isolated singular point since every
deleted E neighborhood of it contains points on the negative real
EXAMPLE 2. The function
sinh z
(0 < lzl < oo)
has a pole of order m = 3 at zo = 0, with residue bl = 1/6.
There remain two extremes, the case in which all of the coefficients in the
principal part are zero and the one in which an infinite number of them a
SEC.
EXAMPLES
67
237
While the theorem in Sec. 66 can be extremely useful, the identification of an
isolated singular point as a pole of a certain order is sometimes done most efficiently
by appealing directly to a Laurent series.
EXAMPLE 4.
If, for insta
10. Recall (Sec. 10) that a point zo is an accumulation point of a set S if each deleted
neighborhood of zo contains at least one point of S. One form of the Bolzano-Weierstrass
theorem can be stated as follows: an in3nite set ofpoints lying in a closed b
follows that all of the coefficients
in the Taylor series for f (z) about zo must be zero. Thus f (z) = 0 in the neighborhood
No, since Taylor series also represents f ( 2 ) in No. This completes the proof.
69. ZEROS AND POLES
The following theorem shows
252
APPLICATIONS RESIDUES
OF
CHAP.
7
value (P.V.) of integral ( 2 ) is the number
00
f ( x ) dx = lim
(3)
provided this single limit exists.
If integral ( 2 ) converges, its Cauchy principal value (3) exists; and that value is
the number to which integral
I
FIGURE 85
and, by identifying the coefficient of l/z in the product on the right here, we find that
Bl = 2. Also, since
when 0 < lz - I < 1, it is clear that R2 = 3. Thus
1
In this example, it is actually simpler to write the integrand as the sum of its
Since the integrand here is even, we know from equations (6) in Sec. 71 and the
statement in italics just prior to them that
EXERCISES
Use residues to evaluate the improper integrals in Exercises 1 through 5.
dx
Ans. n/2.
Ans. n/4.
Ans. n/(2&).
Ans. n/6.
CHAP.
7
74. JORDAN'S LEMMA
In the evaluation of integrals of the type treated in Sec. 73, it is sometimes necessary
to use Jordan's lemma,* which is stated here as a theorem.
Theorem. Suppose that
( i ) a function f (2) is analytic at all points z in the
SEC.
267
75
INDENTED P ~ ~ H S
(b) Show that the value of the integral along the arc C R in part ( a ) tends to zero as R
tends to infinity by obtaining the inequality
/
and then referring to the form (3), Sec. 74, of Jbrdan's inequality.
(c) Use the resu
Math 121: Linear Algebra and Applications Solution Set 2
Posted: Fri. Oct. 17, 2007
Prof. Lydia Bieri
Written by: Luca Candelori
Exercise 1 (1.4/12). Recall that for if and only if statements we need to prove both directions: assuming the statement to the
Math 121: Linear Algebra and Applications Solution Set 3
Posted: Tue. Oct. 30, 2007
Prof. Lydia Bieri
Written by: Luca Candelori
Exercise 1 (1.6/35). (a) Let v + W be a typical element of V /W . Since v V , we can write it as v = a1 u1 + . . . + ak uk + a
Math 121: Linear Algebra and Applications Solution Set 4
Posted: Tue. Nov. 6, 2007
Prof. Lydia Bieri
Written by: Luca Candelori
Exercise 1 (2.1/15). Linearity follows directly from the properties of integrals. Namely
x x x
(f (t) + g (t)dt =
0 0 x 0 f (t)
Math 121: Linear Algebra and Applications Solution Set 5
Posted: Fri. Nov. 15, 2007
Prof. Lydia Bieri
Written by: Luca Candelori
Exercise 1 (2.5/3). (a) This matrix is simply the matrix with respect to of the linear transformation T that maps to . In part
Math 121: Linear Algebra and Applications Solution Set 6
Posted: November 30th
Prof. Lydia Bieri
Written by: Luca Candelori
Exercise 1 (2.6/1d-h, 2.7/1c-g). (a) (d) TRUE. Note that for any nite dimensional V , V = (V ) (e) FALSE. Let V = R3 and consider t
Math 121: Linear Algebra and Applications Solution Set 7
Posted: December 3rd
Prof. Lydia Bieri
Written by: Luca Candelori
Exercise 1 (3.3/4b , 3.3/5). (a) We can rewrite the system as
1 2 1 x1 5 1 1 1 x2 = 1 2 2 1 x3 4
The inverse of the coecient matri
Math 121: Linear Algebra and Applications Solution Set 8
Posted: December 13th
Prof. Lydia Bieri
Written by: Luca Candelori
Exercise 1 (5.2/14(b), 16). (a) We want to diagonalize the matrix A= 8 10 5 7
rst. Its eigenvalues are 3, 2 with corresponding eige
Math 121: Linear Algebra and Applications Solution Set 9
Posted: January 9th
Prof. Lydia Bieri
Written by: Luca Candelori
Exercise 1 (7.1/2b). The eigenvalues are = 4, 1, each with multiplicity 1, with corresponding eigenvectors v1 = (2, 3) and v2 = (1, 1
Harvard University
Midterm 1 for Math 121, Fall 2007
Monday, October 22, 2007 Time allowed: 53 minutes 3 pages (including this one)
Last Name: First Name: Harvard ID Number:
Apart from question 1, you must fully justify your answers. On this exam the full
Harvard University
Midterm 2 for Math 121, Fall 2007
Wednesday, November 21, 2007 Time allowed: 53 minutes 3 pages (including this one)
Last Name: First Name: Harvard ID Number:
Apart from question 1, you must fully justify your answers. On this exam the
Harvard University Solutions Midterm 1 for Math 121, Fall 2007 Monday, October 22, 2007 1. (20 points) a) True b) False c) False d) True e) True 2. (40 points) a) Clearly, the zero vector lies in each of the subsets in (i), (ii). Thus, check if for any tw
Harvard University Solutions Midterm 2 for Math 121, Fall 2007 Wednesday, November 21, 2007 1. (20 points) a) False b) False c) False d) False e) True 2. (50 points) a) (30 points) 2 A=1 2 2 2 2 3 1 1
First, we compute the characteristic polynomial pA ().