164
23.
Exercises
Kt
Therefore, a: (t) g 1.50 (Eh, %) . Also, you get
15mg an +Kw(t)
If K > 0, this implies
If X = 0, then the result requires no proof.
With Gronwalls inequality and the integral de
162 Exercises
is a linear map and so it can be represented by a unique I E W so that
b
(M) = / (mm) d:
for all to. Now what about the fundamental theorem of calculus?
th s s h
(to, %) E i]; (to, z (5)
Benjarnin Van Roy and Kahn Mason 27
[-1.11
Figure 2.5: The vectors [1 0]T and [0 1]T are orthogonal? as are [1 1]T and [1 1]T.
On the other hand, [1 O]T and [1 IF are not orthogonal.
Multiplying bot
Benjarnin Van Roy and Kenn Mason 17
cfw_CL 5
cfw_ll 2
cfw_5, 0
(Cl. -1
(a) (b)
Figure 2.2: (a) A case 1with no solution. (b) A case with an innite number of
solutions.
#7.
s3.
49.
50.
._.-_u\.-.:-n.uu.:-_.-.- -u. ._ ._._. .
Section 1.6
Clearly W1 n W2 is nonempty; it contains 0. Let
v,v E (W1 n H2). Then vtw'e W and v,v E H so v + v E W
l 2 1
and v + v E W2 since W1
Benjarnin Van Roy and Kahn Mason 21
the last K components come from 132, and b E ERMiK . The representation of
a matrix in terms of smaller matrices is called a partition.
We can also partition a matr
maI
Section 1.6
40. with more equations than unknowns, we naively expect that too
many conditions are demanded of the unknowns to permit any
solution. In practice, this is usually the case (see Sectio
Benjamin Van Roy and Kahn Mason 29
T
= y y_2 aTa + aTa
: yTy _ (:ylg
a a
< yTy
which contradicts that fact that yTy is the minimum of f (051, . . . ,oN). It
follows that N + L = M. Cl
2.2.4 Vect
30
Applying Theorems 2.2.6 and 2.2.7, we deduce the following relationships
among the dimensionality of spaces associated with a matrix.
Theorem 2.2.8. For any matrix A E ERMW, dim(C(A)+dim(N(AT) : M
24
[1 1_,-
(a) (b)
Figure 2.4: (a) The vector space spanned by a single vector (:51 E ER3. (b) The
vector space spanned by a1 and a2 E R3 where cfw_12 is not a multiple of a1.
Benjamin Van Roy and Kahn Mason 19
2. 1 .4 Multiplication
A row vector and a column vector can be multiplied if each has the same
number of components. If :13, y E 3%, then the product may of the row
Benjamin Van Roy and Kahn Mason 25
(11,. . . ,aN are linearly independent. Thus I) must be linearly independent of
a2, . . . ,aN , and so the set is linearly independent.
To show that the set spans 5'
Chapter 2
Linear Algebra
Linear algebra is about linear systems of equations and their solutions. As a
simple example, consider the following system of two linear equations with
two unknowns:
| |
p_|
156 Exercises
First show this is actually a norm. Next explain why
_An+l
)ll
lllAXIHEA sup X Eklllxlll-
HEE+
First, it is obvious that there exists A < 1 such that
9e)
Just pick A larger tha
Exercises 165
where D is the diagonal matrix obtained from the eigenvalues of A and N5 is a nilpotent
matrix commuting with D which is very small provided 8 is chosen very small. Now
let 11 (t) be the
Exercises 153
E33 Exercises
14.7
1. Solve the system
4
1
026
using the Gauss Seidel method and the Jacobi method. Check your answer by also
solving it using row operations.
4 1 1 :L" 1
1201
1 5 2 y =
154 Exercises
Show the norm H which comes from an inner product is strictly convex.
Let $.11; be as described.
2
413 + y 33 y 1 2 1 2 2
2 2 = 5 Ha + 5 Hall = use
If m 35
Exercises 157
14.
15.
A matrix A is diagonally dominant if le'l 2;:- 23:73;- |o.,-,-| . Show that the Gauss Seidel
method converges if A is diagonally dominant.
It is the eigenvalues of 310 which are
Exercises 161
HIUl'vaw -(1'(3)Uanl lIOilv- III(3W IWI
11 (t) - 1 (3) I'vl livl
IA IA
and so this is continuous.
1 (t) v - 1 (3) ill S llI (t) - l1(3)| lvl
so t > @(tr is continuous. Differentiable wor
Exercises 155
Say 10 = 23:1 chk. Dene a norm on V by Hts E (221:1 )cklz) U2 where w =
23:1 ckok. This gives a norm on V which is equivalent to the given norm on V.
Thus if it! > w, it follows that c >
Exercises 167
vvhere D : , each Ag, real. Then the above equals
An.
0 (sweat a
Z T = 9
k=o '
Where H : U*DU is obviously Hermitian.
1
30. If U is unitary and does not have 1 as an eigenvalue so that (
Exercises 169
Rather than use this method as described, I will just use the variation of it based on
the QR algorithm which involves raising the matrix to a power which was just used
to nd the rst eig
158
Exercises
Now letting z = u + is where me are real valued, show
u+bgu = 0,u(0)=1,u(0)=0
v+b2e : l]. v(0):0,e(0):b.
Next show a (t) = cos (hi) and e (t) = sin (ht) work in the above and that there
Exercises 159
The series converges absolutely by the ratio test and the observation that Ah g
Auk Now we need to show that you can differentiate it. Writing the difference
quotient gives
where s (k, h
.- Section 1.3
20.
1 -l 2 5
-l h -? 3 .
2 _? -1 6 21. T F F T F T T T F T (e & 1.
5 3 6 3
22. a) Let x be a 1 X n row vector. If 1A is defined,
is n x m far same m, and xA is than a l x m row
b) Simil