Second Order Linear Equations
1. Let y 2 e, so that y 2 re and y = r2 (3. Direct substitution into the
differential equation yields (12 + 27 3)e" m 0. Canceling the exponential, the
characteristic equation is 72 + 27' 3 = O. The roots of the equation are
Exam 1 Math 326-Linear Algebra
February 6, 2014
Name:_
Definitions: (9) Complete the following.
1. A set of vectors cfw_v1,vp in
R
n
is said to be linearly independent if _
_.
2. A transformation (or mapping) T is linear if:
(i) _
(ii) _.
Problems: Answer
Final Review for Linear Algebra
The exam is scheduled for Friday, May 9, 2014, from 8:00 a.m.-10:00 a.m. and will cover the
material from the first four tests, as well as Sections 2.5, 2.6, 3.1-3.5 from Boyce/DiPrima.
*Does not include the reduction of or
Practice Exam 4
Math 326
Name:_
1 1
0 -1
1. Answer each question about the matrix A =
0 1
0 - 1
0
3
.
3
1
a) Do the columns of A form a basis for Col A? An orthogonal basis?
b) Use the Gram-Schmidt process to find an orthogonal basis for Col A.
5
0
c) F
Exam 4 Review
Math 326: Linear Algebra & DEs
The exam is scheduled for Thursday, April 17, 2014, and will cover Sections 6.2-6.6 in Lay and
1.1-1.3, 2.1-2.4 from Boyce and DiPrima.
Definitions: Be able to state and apply!
Orthogonal/orthonormal basis for
Exam 3 Review
Math 326: Linear Algebra
The exam is scheduled for Thursday, March 27, 2014, and will cover Sections 4.2-4.6,
5.1-5.3, 6.1, and part of 6.2 (orthogonal, orthonormal sets only) from the text.
Definitions: Be able to state and apply!
Null spac
Exam 2 Review
Math 326: Linear Algebra
The exam is scheduled for Thursday, February 27, 2014, and will cover Sections 2.2-2.5,
3.1-3.3, 4.1 from the text.
Definitions: Be able to state and apply.
Inverse of a matrix
Invertible Matrix Theorem
Invertible li
Math 326: Linear Algebra
Exam 1 Review
The exam is scheduled for Thursday, February 6, 2014 and covers Sections 1.1-1.9, 2.1 in Lays text.
Definitions:
System of linear equations
Solution of a system of linear equations
Consistent system
Elementary row op
Math 326
Exam 1 Practice Questions
x +y +2z +3w =13
1. Consider the system of equations: x - 2y +z +w =8 .
3x +y +z - w =1
a) Express the system as a vector equation.
b) Express the system as a matrix equation Ax = b.
c) Form the augmented matrix for the
Math 326
Exam 2 Practice Questions
2 3 0 1
1 0
1
4 5 3 3
0 0
1
1. Consider the matrices A =
and B =
.
2 - 6 7 7
2 2
1
-1
a) Compute, if possible: B-1, ( AT A) , ( A AT )
-1
T -1
b) Find (B ) , if it exists.
T
c) Solve the equation (XB) = A for X,
Exam 2 Math 326
Thursday, February 27, 2014
Name:_
Definitions/Theorems: (10) Complete the following.
1. The matrix B is an inverse of the matrix A if _.
2. The Invertible Matrix Theorem: Let A be an n x n matrix. Then the following are equivalent.
(Note:
Exam 3 Math 326
March 27, 2014
Name:_
Definitions/Theorems: (8) Complete the following.
1. The scalar
is an eigenvalue of an m x n matrix A if _
_.
2. Let H be a subspace of the vector space V. The set cfw_b1, , bp in H is a basis for H if
i) _
ii)_ .
Fil
Exam 4 Math 326
April 17, 2014
Name:_
Circle ALL that apply (9):
1. The differential equation
is:
FIRST ORDER
2. The differential equation
LINEAR
LINEAR
SEPARABLE
LINEAR
SEPARABLE
is:
FIRST ORDER
3. The differential equation
SEPARABLE
is:
FIRST ORDER
Fill
Exam 4 Math 326 Name:
Apri117, 2014 J
Circle ALL that apply (9):
1. The differentiaiequatmn ;_(cosx):3i'+:t-y+5y"t+l is:
FIRso/RDER SEPAMBLE
cfw_.x _ . Eng 2 g (:3 I f _' i .f- e
- - - 3 I: 2 _ - . r, = cfw_ve 7 u or i
2. The differential equation x 3
Exam 3 Math 326
March 27, 2014
Definitions/Theorems: (8) Complete the following.
-43 on:
1. The scalar 7t is an eigenvalue of an m amatrix A if A X r A E for
4 J l
gnaw X 1 g 2 . .
2. Let H be asnbsggce of the vector space V. The set cfw_bh .,bp in H is a
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m
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ire
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1 ii
,
x; f , a
Practice Exam 3 Math 326 Name:
2
LetA=3 lit-12
OOOOJ
00 23164]
02,3 C
~320
l . Compute each:
WNW
Mr
T i
v
m.
1
5 "i:
a) det(A)= 3? i '5 b) rank<A>=%L_
O 2 is: a? c
t i \
2 3 3: ~ a , KW, *'
, , .\ fa a 9 W535? g "3 A
y; ~ . a, \ ,gr v
39: e) rank (23"
(7740)
Exam 2 Math 7326 Name: i
Thursday, February 27, 2014
Denitions/Theorems: (10) Complete the following.
@ l. The matrix B is an inverse of the matrix A if g : S '1 : .
2. The Invertible Matrix Theorem: Let A be an rm :1 matrix. Then the following a
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Exam 1 Math 326-Linear Algebra Name:
February 6, 2014
Definitions: (9) Complete the following.
1. A set of vectors cfw_v1,. ,vp in R" is said to be linearly independent if E11- 1e cfw_ UM
_.-: A .3 ,
23; 3 X a i we? X Um =5 an 'iTWKthr-
fa?ff 51; (@007
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1%,
3;.
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3.: t.
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Writing Assignment-Linear Algebra (Math 326)
Topic: Linear algebra provides useful tools in a number of quite diverse fields. The purpose of
the paper is to allow you to fully explore a single application of linear algebra. Choose such an
application and