Math 18.06, Spring 2013
Problem Set #6
April 5, 2013
Problem 1 (5.2, 4). Identify all the nonzero terms in the big formula for the determinants
of the following matrices:
1 0 0 1
1 0 0 2
0 1 1 1
0 3 4 5
,
A=
B=
1 1 0 1
5 4 0 3 .
1 0 0 1
2 0 0 1
For A, we
Problem Set 3 Solutions:
Section 3.2
6.
1 3 5
0 0
0
The free variables are x2 and x3 so the special solutions to Rx = 0 are (3, 1, 0) and (5, 0, 1).
1 3 0
Elimination on B gives the row reduced echelon matrix R =
0 0 1
The free variable is x2 so the speci
18.06 (Spring 14) Problem Set 5
This problem set is due Thursday, March 20, 2014 by 4pm in E17-131. The problems are
out of the 4th edition of the textbook. This homework has 8 questions worth 100 points
in total. Please WRITE NEATLY. You may discuss with
18.06 Problem Set 2
SOLUTIONS TO SELECTED PROBLEMS
1. Section 2.5, Problem 25
3 1 1
Answer: A1 = 1 1 3 1 ; the columns of B add up to 0, so B 1
4
1 1 3
does not exist.
2. Section 2.5, Problem 30
Answer: not invertible for c = 7 (equal columns), c = 2 (equ
18.06 (Spring 14) Problem Set 4
This problem set is due Thursday, Mar. 13, 2014 by 4pm in E17-131. The problems are
out of the 4th edition of the textbook. This homework has 8 questions worth 100 points
in total. Please WRITE NEATLY. You may discuss with
18.06 Problem Set 4
SOLUTIONS
1. Section 3.5, Problem 42
Solution: If the 5 by 5 matrix [A b] is invertible, b is not a combination of the
columns of A. If [A b] is singular, and the 4 columns of A are independent,
b is a combination of those columns.
2.
18.06 Problem Set 1
SOLUTIONS
1. Section 2.1, Problem 10
Answer: Ax = (18, 5, 0); Ax = (3, 4, 5, 5).
2. (a) What two vectors are obtained by rotating the plane vectors
1
0
and
0
1
by 30 degrees in the clockwise direction? Write a matrix A such that for
ev
J =
2
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
,
B ~ w k v fe
yY#B h ggWdS
9 d Bc b ` X V T R PH F D B
W GQ)SP aYW @T USQIGEC' @
@
A
P rxH rQH
F ` wcH d
P Q)xwxYSyxWrQH
F Bc b w Hc w R P o wH y
wD
Yy
P AP T
30
04
@ GIIQW8B x#ExSiGyGI#B xEb
P iR
) a p
=4
A
P B d e R B H B
rEXAfGGp QH
F P eH t B R P d R P
S)XS)S"S`
@
p
4 4 8
A = 4 4 8
8 8 16
24, 0, 0 det A = 0
Ba
uitr pi g
qsq)hf
1
A
e Bd c a Y W T R PH F D B @
xGQ)SP b`ws@ USQIGEC' w
@ d RH
a x)A)IQH
nR
X
12
2 13
AT A =
65
56
p B FH R
k =1
`
IW
A
l gi ffXR
A x ed
e Ad c a Y W S Q IG E C A 9
@U FP(RI b`XV9 TRPHFDB& X
1
i
Atk
=I+
k!
k =1
tk
k!
A 3 = A4 = = A
1
i
+
1
2
1
i
=
cos t
sin t
A = I + (et 1)A =
et 3(et 1)
0
1
.
Q W
Ak tk
=I+
k!
+ 1 eit
2
1
2
9
k =1
=I+
1
i
=
9
e
At
A2 = A
1
U be)guI sqyV ige7b`D' s
x Xw v d t r h f dc a Y X V
j
j
V dg j `g "`X7add7agin9b)gnc r dw`g
Xv a
XY
X d v fc jX v d a t X u t d f
d XY
fx)ng`YX7aa$dgdso d a t
hw d d w f
d Y ww h X a Y f
dX`g en)fw7X`nc' V dd`Yg p)ng`7ns`Gnd9ec
X f x hw d Y X a fc w
18.06 Spring 2013 Problem Set 9 Solutions
1. 6.6 #4: Since A has two distinct eigenvalues, it must have 2 linearly independent
eigenvectors. Since A is 2 by 2, this means that A is diagonalizable: A = SS 1
1 0
. It follows that if B is 2 by 2 and has the
Pset #2 Solutions
2.6 #13 We can guess the following decomposition:
L=[1000
1100
1110
1111]
U=[a a
0 b-a
0 0
0 0
a
a
b-a b-a
c-b c-b
0
d-c ]
2.6, #24 Suppose A=LU for an invertible A; then A_k = L_k U_k, where A_k denotes the k x k matrix
formed by lookin
Course 18.06: Problem Set 4
Due 4PM, Thursday 15th October 2015, in the boxes at E17-131.
This homework has 5 questions to hand-in. Write down all details of your solutions, not just the
answers. Show your reasoning. Please staple the pages together and c
Course 18.06: Problem Set 3
Due 4PM, Thursday 8th October 2015, in the boxes at E17-131.
This homework has 4 questions to hand-in. Write down all details of your solutions, not just the
answers. Show your reasoning. Please staple the pages together and cl
Course 18.06: Problem Set 2
Due 4PM, Thursday 24th September 2015, in the boxes at E17-131.
This homework has 4 questions to hand-in. Write down all details of your solutions, not just the
answers. Show your reasoning. Please staple the pages together and
Course 18.06: Problem Set 1
Due 4PM, Thursday 17th September 2015, in the boxes at E17-131.
This homework has 4 questions to hand-in. Write down all details of your solutions, not just the
answers. Show your reasoning. Please staple the pages together and
We Mr [Amie a Coimwg gwumzndah I,
Vkeo a" ENLUGS non *4:ijth
53+c-L +c 7/0, i «20
Z
;.(.
Kg
3
-J. ; J.
gécég. (6A+
C) A colvmvx {S {tee H is 0 (,MW (OMLHGAIM
d Hre column Lafore,'4. Wm M GA!) «We for
1 $154 cot/w» :4 ;.g 0 <=> 2:0.
A; we Saw {1 {014 a}
4.1
SOLUTIONS
Notes: This section is designed to avoid the standard exercises in which a student is asked to check ten axioms on an array of sets. Theorem 1 provides the main homework tool in this section for showing that a set is a subspace. Students sho
Math 18.06, Spring 2013
Problem Set #Exam 3
May 14, 2013
Problem 1. In all of this problem, the 3 by 3 matrix A has eigenvalues 1 , 2 , 3 with
independent eigenvectors x1 , x2 , x3 .
a) What are the trace of A and the determinant of A?
The trace of A is 1
18.06 Homework 7
Professor Strang
due April 17, 2014
This problem set is due Thursday, April 17, 2014 by 4pm in E17-131. This homework
has 9 questions worth 90 points in total. Please WRITE NEATLY. You may discuss with
others (and your TA), but you must t