Eigen Problems and Diagonalization Using Matlab
An Eigenproblem for a given matrix A requires nding the set of vectors, x, and the scalar numbers such that Ax = x. In other words, we want the vectors which, when operated on by A, are simply multiples of t
ll.
13.
15‘
17.
9.2 Orthononnal Bases I97
Next. calculate l/(ai + l) i U- The matrix
4+ 1 WT : (~3it/j)/6 (m3+ ﬁve
a|+l (—3+\/3)/6 Ham/6
Finally, we put all the part together to get the matrix
i/x/i —1/~/§ 1N5
A: —i/\/§ (~3—ﬂlf6 (—3+\/§)/6
I/ﬁ (—3+\/§)/
MATH 107.01
HOMEWORK #17 SOLUTIONS
Problem 5.3.1. Let T : R2 R2 be the linear transformation
x1
x1 + x2
T
=
.
x2
x1 x2
(a) Find [T ] where is the standard basis of R2 .
(b) Let be the basis for R2 consisting of
1
2
,
.
1
1
Find the change of basis mat
MATH 107.01
HOMEWORK #12 SOLUTIONS
Problem 4.2.2. Determine the general solution to the differential equation
y 00 y 0 6y = 0.
Solution. The characteristic polynomial is
p() = 2 6 = ( 3) ( + 2) .
Corollary 4.7 in the book then implies that the general sol
MATH 107.01
HOMEWORK #13 SOLUTIONS
Problem 4.3.1. Determine the general solution of
y 00 y 0 6y = 3e2x .
(1)
Solution. The characteristic polynomial is
p() = 2 6 = ( 3) ( + 2)
so that the homogeneous solution is
yH = c1 e3x + c2 e2x .
By Theorem 4.11, the
MATH 107.01
HOMEWORK #7 SOLUTIONS
Problem 2.2.1. Determine which of the following sets of vectors are subspaces of
R2 .
x
(c) All vectors of the form
.
2 5x
x
(d ) All vectors of the form
where x + y = 0.
y
0
Solution. (c) This is not a subspace since t
MATH 107.01
HOMEWORK #19 SOLUTIONS
Problem 5.5.5. Diagonalize
A=
0
4
3
0
if possible.
Solution. From Homework #18, the eigenvalues of A are 1 = 2 3 and 2 = 2 3.
Furthermore, the eigenspaces of A are given by
3/2
3/2
, E23 = Span
.
E23 = Span
1
1
It fol
MATH 107.01
HOMEWORK #11 SOLUTIONS
Problem 4.1.2. Determine if the differential equation
ey y 00 4y 0 + 5xy = 0
(1)
is linear or not. If it is linear, then give its order.
Solution. This is not linear.
Problem 4.1.3. Determine if the differential equation
MATH 107.01
HOMEWORK #16 SOLUTIONS
Problem 5.2.6. Let S, T : R2 R2 be the linear transformations
x
2x y
x
x + 3y
S
=
, T
=
.
y
x + 2y
y
xy
Find
TS
x
.
y
Solution. Compute
x
2x y
(2x y) + 3 (x + 2y)
5x 5y
TS
=T
=
=
.
y
x + 2y
(2x y) (x + 2y)
x
Lecture Schedule, Math 216, 2016-2017 Fall, Clark Bray
Use the "Lecture #" below and the Recordings Schedule to identify which recordings you should
watch before a given class meeting.
Syllabus exercises are assigned by weekly email from the instructor
Th
Math 216 Syllabus
Topic(s)
Systems of Linear Equations
Matrices and Matrix Operations
Inverses of Matrices
Special Matrices
Determinants
Further Props. of Dets.
Proofs on Dets.
Linear Independence in n
Vector Spaces
Subspaces
Linear Independence
Dimension
MATH 107.01
HOMEWORK #14 SOLUTIONS
Problem 3.6.7. A 100-gal tank contains 40 gal of an alcohol-water solution 2 gal
of which is alcohol. A solution containing 0.5 gal alcohol/gal runs into the tank
at a rate of 3 gal/min and the well-stirred mixture leave
MATH 107.01
HOMEWORK #15 SOLUTIONS
Problem 5.1.3. Determine if T : R3 R3 given by
x
x+y+z
T y = z y x
z
xyz
is linear.
Solution. Note that
1
0
2
2
4
4
T 0 + T 1 = 0 + 0 = 0 6= 0 = T
1
1
0
0
0
2
1
1
0
1 = T 0 + 1.
2
1
1
Hence T is not linear.
3
2
MATH 107.01
HOMEWORK #18 SOLUTIONS
Problem 5.4.5. Find the eigenvalues and bases for the eigenspaces of
0 3
A=
.
4 0
Solution. The characteristic polynomial is
3
pA () = det (I A) = det
= 2 12 = 2 3 + 2 3 .
4
The eigenvalues of A are then 1 = 2 3 an
Math 216 - Braley
Problem Set 4
DUE: Monday, 6/15/15
1. Let T : V W be a linear transformation. Show that the image of a basis of V
completely determines all values of T.
2. Let T : V W be a linear transformation. Show that T is one-to-one, if and only if
Math 216 Problem Set 2
Justin Xu
Due: May 29 2015
1. Can R3 contain a 4 dimensional subspace? Explain.
No, R3 cannot contain a 4 dimensional subspace. Although we can form a subspace W in 4
dimensions that appears to follow the necessary additive and scal
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Math 216 Problem Set 1
Justin Xu
Due: May 22, 2015
1. Determine if the homogeneous system of linear equations has a nontrivial solution:
xy+z =0
2x + y 2z = 0
3x 5y + 3z = 0
We begin by re-writing the above system as an augmented matrix:
1 1 1 0
2 1 2 0
Math 216
2015-2016 Spring
Additional Homework Problems
1. In parts (a) and (b) assume that the given system is consistent. For each system determine all
possibilities for the numbers r and n r where r is the number of nonzero rows of the (reduced) row
ech