Exam 4 Review Guide Blank real
of 7
Transcript of Exam 4 Review Guide Blank real
-
8/14/2019 Exam 4 Review Guide Blank real
1/7
Review Guide for Exam 4, Page 1 of 7
REVIEW GUIDE FOR EXAM 4
SECTIONS 7.17.6, 7.8, 7.9
1. a) Set up an augmented matrix for the system
25
242
682
43
421
41
xx
xxx
xx
and completely reduce the
augmented matrix using exactly two row operations.
b) Express the infinitely many solutions as a linear combination of vector(s).
2. Interpret the reduced matricesa)
1
0
0
0
2
1
0
1
0
0
0
1
b)0
0
0
0
0
1
0
0
0
0
0
0
-
8/14/2019 Exam 4 Review Guide Blank real
2/7
Review Guide for Exam 4, Page 2 of 7
3. Circle the vector,t
t
e
et
3
3
)((1)
x ort
t
e
et)(
(2)x , that is a solution of xx'
12
21)(t .
4. Given the equation c1v(1)
+ c2v(2)
+ c3v(3)
= 0.
a) By definition the vectors v(1)
, v(2)
, and v(3)
are linearly independenton some intervalIif(circle one):
i. This equation is true onIfor nonzero values ofc1, c2, orc3.ii. This equation is true onIif and only if c1 = c2 = c3 = 0.
b) Find the relationship of dependency for the vectors:
v(1)
=
3
2
1
, v(2)
=
1
1
0
, v(3)
=
2
0
2
-
8/14/2019 Exam 4 Review Guide Blank real
3/7
Review Guide for Exam 4, Page 3 of 7
5. Convert the third order initial value problem, txxxx 34'4''''' , x(0) = 1,x (0) = 2,x (0) = 3,into a corresponding system. Write the system in matrix-vector form.
6. The equation that defines ras an eigenvalue of the matrix A with corresponding eigenvector is
7. Find the general solution of the system xx3324D . Express the general solution as a linear
combination of solution vector, in terms of (t), and as a system of algebraic equations.
-
8/14/2019 Exam 4 Review Guide Blank real
4/7
Review Guide for Exam 4, Page 4 of 7
8. The eigenvalues of a third order system are 41r , 12r , 13r , with corresponding eigenvectors
0
0
1
(1) ,
2
1
0
(2) ,
7
5
0
(3) . If the initial condition is
1
5
1
)0(x , find the specific solution.
9. Find the real-valued general solution ofx = x. Express the general solution as a linearcombination of solution vectors.
-
8/14/2019 Exam 4 Review Guide Blank real
5/7
Review Guide for Exam 4, Page 5 of 7
10. Find the general solution ofx = x. Express the general solution as a system of algebraicequation
11. Consider the system . The fundamental matrix of the correspondinghomogeneous system is (t) = . Find the particular solution X(t) of the non-homogeneous
system and express as a single vector.
-
8/14/2019 Exam 4 Review Guide Blank real
6/7
Review Guide for Exam 4, Page 6 of 7
12. Tanks A and B are interconnected. Tank A initially contains 15 lbs. of salt dissolved in 100 gallons
of water. Tank B initially contains 100 gallons of pure water. Pure water flows into tank A at therate of 3 gal/min. A brine solution of lb of salt per gallon flows into Tank B at the rate of 1
gal/min. Brine flows from tank A to tank B at a rate of 4 gal/min. Brine flows from tank B to tank
A at a rate of 1 gal/min. Brine leaves the system from tank B at a rate of 5 gal/min. Letx1(t) and
x2(t) denote the amounts of salt in tanks A and B, respectively, at any time t. Set up a system ofequations and initial vector to model this system.
-
8/14/2019 Exam 4 Review Guide Blank real
7/7
Review Guide for Exam 4, Page 7 of 7
13. An extra problem for practice: Find the general solution of the system
3213
3212
3211
'
'
3'
xxxx
xxxx
xxxx
