Solutions to Review Problems for Exam 1.
Chapter 1
A.
7. (H S ) P or P (H S )
8. H S P or (H S ) P
11. (S H ) P
12. O V (G L) or O V G L
13. N (M L)
14. C R P
Chapter 2
E.
3. Not Valid.
O
F
M
F
G
F
V
F
OM
T
O
T
G
F
V
F
GV
T
M V
T
G M
T
O G
F
7. Not Valid.
Chapter 6
The parts of the interpretations not specied can be dened arbitrarily.
A. 10. (x)(y )[P x Qy (z )(Rxy Sxyz )];
S = U, P, Q, R, S
a
Q
F
a
true: U = cfw_a
P
F
P
T
Q
T
false: U = cfw_a
R
a
11. (x)(y )[P x Ry Sxy (z )(P x Rx Qxaz )];
true: U = cfw_a
MA 262
Solution for Exam 1
Spring 2013
1. a) Solve the following initial-value problem:
dy
+ 3ty = 2t,
dt
y (0) = 1.
This equation is linear (and separable), with integrating factor
I = exp
3t dt
= exp(3t2 /2).
Therefore, we get
(Iy ) = 2t exp(3t2 /2) Iy
MA 262 PRACTICE PROBLEMS
page 1/8
Circle the letter corresponding to your choice of correct answer.
1. Find the determinant of A if
1
1
A=
1
0
1
2
2
3
1
2
3
4
2
0
0
1
A. 1
B. 0
C. 3
D. 7
E. 11
p1
q1
p
q
2. If A is a 3 4 matrix and P = 2 and Q = 2 are
p3
Study Guide for MA 262
Linear Algebra and Dierential Equations
Topics covered in the exam
Linear Algebra:
1. Complex numbers, polar representation, roots of complex numbers.
2. Vector spaces: subspace, basis, spanning set, linear combination, linear indep
1
Solutions to Section 1.1
True-False Review:
1. FALSE. A derivative must involve some derivative of the function y = f (x), not necessarily the rst
derivative.
2. TRUE. The initial conditions accompanying a dierential equation consist of the values of y,
i
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CHAPTER
3
Determinants
Mathematics is the gate and key to the sciences. Roger Bacon
In this chapter, we introduce a basic tool in applied mathematics, namely the determinant of a square matrix. The determinant is a number,
The inverse square matrix
Exam 1: Lessons 1-14 (recitation in two weeks)
Sometimes we are in the situation that for a given n x n matrix A, there
exists a matrix B satisfying
Questions
1. When does such a matrix B exist?
2. If B exists, how do we nd it?
Cofactors Expansion Part 2
Let A be an n x n matrix throughout
Each cofactor C Is the determinant of (n - 1) x (n - 1) matrix.
Build a matrix of cofactors, M , the (I,j) entry will be equal to C of the
original matrix A.
The ad joint matrix of A wr
Gaussian Elimination
Ex: Solve the system of linear equations
Names
1. If we put augmented matrix A into REF and then use back substitution
called Gaussian elimination.
2. Put matrix A into RREF. No longer need to use substitution. This is
called gau
Properties of determinants
Fact: Let A be an n x n matrix. If A is upper triangular than the
determinant of A is - det(A) = a11a22a33ann
Elementary row operations and determinants
Let A be the original. Let B be the result of the operation:
Fact:
Let A
Cofactor Expansion Part 1
Let A be an n x n matrix.
The minor, M , of an entry is the determinant of the matrix obtained by
deleting the I row and j column of A.
A vector space is a set of objects (vectors) that sum up all that we know
about them.
Theorem: Let S be a non empty subset of a vector space V. Then S is a
subspace if and only if S is closed under addition and scalar
multiplication.
The idea is that U3 is redundant in list 3 - it is already a linear combination
of U1 and U2 and throwing it away does not change the span.
Fact: any linearly dependent set of vectors contains a linearly
independent subset with the same span.
Minimal s
Elementary row operations (ERO's) and row echelon matrices
Given a m x n system of linear equations. Let A be the augmented matrix
of the system.
Operations
1. Permute the equations
2. Multiply an equation by a non zero constant
3. Add a multiple of one
Two types of problems that can be turned into a separable or a rst order
DE problem by changing variables.
Type 1: homogeneous DE
We will be able to make this into a separable DE
Type 2: Bernoulli
We will be able to make this into a rst order DE
Type 1:
MATH 262 Quiz 1
\
Name: 5 S a M m PID: Section:
Solve the problem systematically and neatly and ShOW all your work.
> 233%: = WM WW :7: .
=> 0 3- 95 2" Bavl
X -
(3ptsl 2. Do you think that the initial value problem
2: = msin(:z:+y),y(0) = 1
has a unique s
a3:
MAT 262 \ Quiz 2
5;, W
Name: . PH: 7 Saction;
Soi've the problem sygtematimy am neatly and ShOW all ycmr work.
W
(13pm) 1.801%: the differential equation
_J_
PM): xmx d 4~$my mgm?
.j_ 5
1m: 65" x lhx fgxl/hxdx
ol d z * :I Y 7"de JV >V 3X
aw' 9" M
MATH 1262 Quiz 3
SW
Solve the pmMem syswll'laticrval1y and neatly and Show 2111 your WUik.
2
(3pm) 1. Solve the initial value problem Qf 7 ) = y 1-, 1') f(>l) : +5.
2.1:? cfw_ii? 2* (.712 i 22/ 4*, 00!?) m 0, 90) L 2
J . .
& M53X7/
N" \(l/y-Hl ax:
szm: P
MATH 262 Quiz 4
Name: SW PID: Section:
Solve the problem systematically and neatly and Show all your work.
(3pts) 1. Prove that a symmetric upper triangular matrix is diagonal.
Pf . A : CR;JJ - 51er Wx
A 5 53WMC =5 T 3 ? :3 (VJ=0 mu tia
AAWW 5) 49:0 W L>J
MATH 262
j \ Quiz 9 ;
:8 ll PID: a ' i
Name: Section:
Solve the problem systematically and neatly and show all your work.
USNJII
1. (2pts) Let T : R2 > R3 be the linear transformation for which A
l
TwpH T<ll) lg T<H>C
54 [Ll liljr U]? C'+:7:l c
2% 7(a):
MATH 262 Quiz 10
Name: 539! m PID:
Solve the problem systematically and neatly and show all your work.
1. (2pts) An annihilator of cos :1: + arem + 1 is
(a) D<D 1)(D+1> w * DH
(b) (D2_1)(D+1) Mex ~._ (D4)
(0)13 ~11<D2+ 1)
D(D 1)2(D2+ 1) l . D
(e) D(D1)(D2
MATH 262
8 Mm" PEI);
Soive the problem systematicaily and neay and Show all yaur work.
Name :
1. (2 pm) If A and B 3% 3 X 3 matrices such that (EMA) 2: -3 and (1911(8) 3: 2, nd
ciet(A(w23).
g9 Mmrm -= (aficmdww = ng (4) x z cfw_cg
W Mm = Wm
obit/m): WWW
MATH 262 W Quiz 8
Name: PID:
Solve the problem systematically and neatly and show all your work.
1- (213138) Which of the following sets of vectors forms a basis for R3 (2
-" 1
H l l l l
0 0
i, H H '" bfvw V: 42 W3 ,thoml
1 3
OPH
J (14,0th Wat yam
A
9?
v