Optimization
Review
1. Write out the second order Taylor expansion for
2
2
f (x) = ex1 +x2 + 2x1 x2
for
x0 =
1
0
.
2. Use the chain rule to find Df (x) for
f (x) = g(h(x)
where
h(x) =
sin(x2 )
cos(x1 )
,
and
g(y) = y14 + y22 .
3. Perform 2 steps of golden
2.3.
High-dimensional linear systems. Invariant subspaces
hence we obtain
eAt =
where
eAj t
0
eA1 t
eA2 t
.
.
0
eAm t
39
,
t
for Aj = ()
e
2
2
3
Re et Im et 3
cos(t) sin(t)
t
=
e
= Im et Re et
sin(t) cos(t)
2
3
for Aj =
.
So, the general solution (2
TIME-DOMAIN ANALYSIS OF A CONTINUOUS-TIME SYSTEM
An Example using The Classical Method
THE PROBLEM
Solve the system equation:
If the input is:
And the initial conditions are:
( D 2 + 3D + 2) y (t ) = Df (t )
f (t ) = t 2 + 5t + 3
y (0) = 2 and Dy (0) = 3
MATH-3435, Fall-2015
Quiz #2
Your name:
MATH-3435 Linear Algebra I CRN 80517
Quiz 2 Sections 1.3-1.4
Directions: Do the quiz on your own paper and attach to this quiz assignment sheet. Clearly
show all your work. For full credit, all your answers must be
MATH-3435, Fall-2015
Quiz #9
Your name:
MATH-3435 Linear Algebra I CRN 80517
Quiz 9 Sections 4.3-4.4
Directions: Do the quiz on your own paper and attach to this quiz assignment sheet. Clearly
show all your work. For full credit, all your answers must be
MATH-3435, Fall-2015
Quiz #7
Your name:
MATH-3435 Linear Algebra I CRN 80517
Quiz 7 Sections 3.1-3.2
Directions: Do the quiz on your own paper and attach to this quiz assignment sheet. Clearly
show all your work. For full credit, all your answers must be
MATH-3435, Fall-2015
Quiz #5
Your name:
MATH-3435 Linear Algebra I CRN 80517
Quiz 5 Sections 1.8-1.9
Directions: Do the quiz on your own paper and attach to this quiz assignment sheet. Clearly
show all your work. For full credit, all your answers must be
MATH-3435, Fall-2015
Quiz #4
Your name:
MATH-3435 Linear Algebra I CRN 80517
Quiz 4 Section 1.7
Directions: Do the quiz on your own paper and attach to this quiz assignment sheet. Clearly
show all your work. For full credit, all your answers must be justi
MATH-3435, Fall-2015
Quiz #1
Your name:
MATH-3435 Linear Algebra I CRN 80517
Quiz 1 Sections 1.1-1.2
Directions: Do the quiz on your own paper and attach to this quiz assignment sheet. Clearly
show all your work. For full credit, all your answers must be
®F blank Let B be a basis for R. As usual, if V 6 RH denote by [v]B the coordinate vector of v with respect to
the basis B. Ifx and y are in R11 and if [x]B = [y]B, then x MUST equal y.
6 F blank Let B be a basis for R. As usual, if V 6 RH denote by [V]B
1) Let 7~ : R2 ~+ R2 be a linear transformation that maps a : (3,7) into (4.1) and
maps v =(3,0) into (4.8)V Use the fact that Tis Iinearto ne the images under Tof
a) N222) ;
mm: arm): (Rom: (my
b) TU} 3{~)
} I
A
,f.
3
¢
03
A
,1:
ca
\J
P7
0Q
K.)
(
SQ
O)
[3:31. Fin an LU factorization for | 2 3 5 l
0 I l I
l we 0 '37 Sf ¢,:s/ns/¢¢r7/2¢y;¢3%¢f
)- 1
a F A
0/6" s 3 3 w 21' _.
- I I: Z 7
M I W 4/7. J L 0 ILLILLM 0 Lg/it a
y; 0 z
o
I a) Find det A.
| b) Find the eigenvalues of A.
| c) Is A invertible? E
Let A and B be m-xn matrices and assume that A is row equivalent to B. Then:
(i) F blank
® F blank
T® blank
® F blank
T ® blank
(D F blank
row space A MUSTequalrow space B.
rowspace A MIGHT equal row space B.
column space A MUST equal column space B.
colu
HH
'TJTJ
'%
1'11
TF
F
blank
blank
blank
blank
blank
blank
blank
blank
If the {V1, V2, V3} are independent and W is a 4th vector, then {V1, v2, V3, W} IVIUST be independent.
If the {V1, V2, V3} are independent and W is a 4th vector, then {V1, V2, V3, W} MI
1 2 7 8 9
1. Let A: 2 3 5
3 4 7 10 11
column space of A. (6 points)
7 8 . Determine the dimension of the null space of A and the dimension of the
78912789 12789
578:019 9 10:.>019910
12
A22 3
3471011 02 141416 00 4 4 4
Since there are two free variables t
. Find the determinant of the matrix
1 2
3 4
. (continued) The (1,2)th element of the inverse of the above matrix is
1 / 5
10
. Find the determinant of the matrix
1 2 2 4
1 2 3 0
0 0 71 2
1 2 3 4
78
. The determinant of the following
cow?) ;, CZ <x.)+ ( 7.)]; C x,+t1, w
\_/ YT. yl-r ykfIh " t
a
(up C(xyupg) we im
u. . in I! is in,
aw am wad ~
5!? Chm): (z «47 I5 G/Wv-é/J .
3' a A da I 7 #777 i\
CC x rvlahm cg Q09 ctr/.4 [ac Vl-éwCof a5. a (tll'tlfigfgcéWlé
faiaxltc/m a( 9 : _400 9-
MATH-3435, Fall-2015
Quiz #10
Your name:
MATH-3435 Linear Algebra I CRN 80517
Quiz 10 Sections 4.5-4.6
Directions: Do the quiz on your own paper and attach to this quiz assignment sheet. Clearly
show all your work. For full credit, all your answers must b