ALMA ROCHA
Assignment Spring 2016 Set 10 due 04/16/2016 at 09:00am EDT
R3
1 8 9 2
-2 -2 -4 3
4. (1 point) Let A =
-2 -2 -4 3 .
-2 -2 -4 3
Find
a
basis
of
the
image
of
T where T (x) = Ax.
R3
1. (1 point) Suppose T :
is a linear map such that
!
!
1
-3 9
-5 15
1. (1 point) Let A =
-3 9 .
-2 6
nullspace(A) is a subspace of Rk where k is
colspace(A) is a subspace of Rm where m is
Let B = 1 3 0 4 -6 .
rowspace(B) is a subspace of Rk where k is
nullspace(B) is a subspace of Rm where m is
6 3 0
3 0 2
ALMA ROCHA
Assignment Spring 2016 Set 02 due 02/06/2016 at 09:00am EST
6. (1 point) Solve the separable differential equation
p
dy
10x 2y x2 + 1 = 0.
dx
Subject to the initial condition: y(0) = 2.
y=
.
1. (1 point) Which of the following are separable dif
ALMA ROCHA
Assignment Spring 2016 Set 01 due 01/30/2016 at 09:00am EST
you can explain why this is true.
1. (1 point) It can be helpful to classify a differential equation, so that we can predict the techniques that might help us to
find a function which
ALMA ROCHA
Assignment Spring 2016 Set 12 due 04/30/2016 at 09:00am EDT
spring16mth165
(b) With Q(0) = 5 and Q0 (0) = 0, we have A = 5 and B = 52 ,
so that Q(t) = (5 + 25 t)et/2 .
1. (1 point) This problem concerns the electric circuit shown
in the figure
ALMA ROCHA
Assignment Spring 2016 Set 07 due 03/19/2016 at 09:00am EDT
6
6
and ~y =
. Find the
1
4
vector ~v = 5~x 6~y and its additive inverse.
~v =
~v =
5. (1 point) Determine if each of the following sets is a subspace of Pn (R), for an appropriate val
ALMA ROCHA
Assignment Spring 2016 Set 04 due 02/20/2016 at 09:00am EST
3. (1 point) Given the augmented matrix
1
1
4 1
4
3
1 ,
A= 3
2 5 6
1
1. (1 point) Write the augmented matrix of the system
+4z=2
70x
6x 8y z= 4
48x+48y
=8
perform each row operation i
ALMA ROCHA
Assignment Spring 2016 Set 08 due 03/26/2016 at 09:00am EDT
5. (1 point) Determine which of the following pairs of functions are linearly independent. Note: For questions involving
trig. functions, the identities cos(3) = 4 cos3 () 3 cos() and
ALMA ROCHA
Assignment Spring 2016 Set 06 due 03/05/2016 at 09:00am EST
1. (1 point) Find the determinant of the n n matrix A with
5s on the diagonal, 1s above the diagonal, and 0s below the
diagonal.
det(A) =
spring16mth165
5. (1 point) Determine all mino
ALMA ROCHA
Assignment Spring 2016 Set 03 due 02/13/2016 at 09:00am EST
spring16mth165
1. (1 point) Suppose the population P of rainbow trout in a
fish hatchery is modeled by the differential equation
dP
= P(6 P),
dt
where P is measured in thousands of tro
ALMA ROCHA
Assignment Spring 2016 Set 11 due 04/23/2016 at 09:00am EDT
1. (1 point) The general solution to the second-order differential equation y00 + 5y0 24y = 0 is in the form y(x) =
c1 er1 x + c2 er2 x . Find the values of r1 and r2 .
Answer: r1 =
an
ALMA ROCHA
Assignment Spring 2016 Set 05 due 02/27/2016 at 09:00am EST
1. (1 point) The inverse of the matrix
5 3
A=
1 6
spring16mth165
3
3. (1 point) If A = 1
1
A1 =
is
A
1
=
b11
b21
b12
b22
,
b21 =
,
b22 =
,
.
,
2
2
1
6
5
3
1
1
0
4. (1 point) Find a
Transpose of a matrix: interchanfe the rows and columns
(A^t)^t=A
(A+B)^t=B^t+A^t
(AB)^T=B^TA^T
Upper and lower triangular matrices.
Upper triangle matrix: is a square matrix where all entries below the diagonal are zeros.
Ex:
1*
02*
003*
Where * can be a
In Homework 14, Part II. A (d), we observed that the eigenvalue with eigenvector ~v is real: hT(~v ), ~v i =
h~v , ~v i = h~v , ~v i, and that hT(~v ), ~v i = hT(~v ), ~v i. So, now suppose the field is complex. Then hT(~v ), ~v i =
h~v , ~v i = h~v , ~v
And many more if we rearrange the order of the Jordan Blocks, but these are the possible Jordan canonical
forms.
(c) If ker(A) and ker(A 2I) are both 2-dimensional, what is the Jordan canonical form of A?
Using Rank-Nullity Theorem, we use dim(ker(T I)d+1
Because 1 and 2 are distinct, 1 6= 2 but 1 h~v1 , ~v2 i = 2 h~v1 , ~v2 i. The statement can only be true if
h~v1 , ~v2 i = 0. Therefore, ~v1 and ~v2 are orthogonal.
(e) Show that V has an orthonormal basis consisting of eigenvectors of T
We use the Gram-S
(b) If J is a matrix in Jordan canonical form, show that J is similar to its transpose.
Now let Bi be the corresponding matrix to Ji such that Pi1 Ji Pi = JiT , with Ji being the corresponding
Jordan blocks to J the Jordan canonical form.
P1
J1
P2
J2
P =
MTH 235: Linear Algebra
Midterm 2
April 4, 2017
NAME (please print legibly):
Your University ID Number:
Circle your instructor:
Evan Dummit
Carl Mc Tague
The presence of electronic devices (including calculators), books, or formula
cards/sheets at this e
Math 235, Spring 2017 (Dummit/Mc Tague) Linear Algebra Midterm 1 Solutions
These solutions are not intended to be exhaustive, but they should be sucient for anyone who has already tried
to work through the problems. Some problems may have multiple dierent
MTH 235: Linear Algebra
Midterm 1
February 28, 2017
NAME (please print legibly):
Your University ID Number:
Circle your instructor:
Evan Dummit
Carl Mc Tague
The presence of electronic devices (including calculators), books, or formula
cards/sheets at th
Math 235, Spring 2017 (Dummit/Mc Tague) Linear Algebra Midterm 2 Solutions
These solutions are not intended to be exhaustive, but they should be sucient for anyone who has already tried
to work through the problems. Some problems may have multiple dierent
MTH 235 Spring 2017 Dummit / Mc Tague Midterm 2 Review Problems
1. Answer the following true/false questions, where T : V W is a linear transformation.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(T/F) The vector v = ex 2ex is in the kernel of the
MTH 235 Spring 2017
c
Dummit / M Tague
Final Part B Review Problems
1. Answer the following true/false questions:
E is the reduced row-echelon form of the matrix A, then rank(E) = rank(A).
(T/F) If A is an m n matrix of rank r , then the solution space of
DAE 1
Lecture One (1.18.17)
1. Modern biology is organized into hierarchal levels
a. Emergent properties emerge at each level
i. Result of interactions among component parts
ii. Properties you couldnt predict by just knowing antecedent
1. Ex. You couldnt
I.
II.
III.
Amphibians
a. Aquatic larval stage
i. Earliest vertebrates released eggs into water
b. Advantages of an egg
i. Gelatinous coating: buffers environment; helps maintain homeostasis
ii. Yolk: provides nutrients
iii. Embryo doesnt dry out
c. Requi
Sensory system in animals
1. Whats ionotropic and metabotropic?
2. Whats the pineal gland?
3. Know how to label the human eye
a. Whats the fovea and blind spot?
b. Whats the retina? What are the three major layers rods and cones? Understand
bipolar and ga
Introduction to Probability Lecture Notes
Version July 28, 2016
David F. Anderson
Timo Seppalainen
Benedek Valko
c Copyright 2016 David F. Anderson, Timo Seppalainen and Benedek Valko
Contents
Preface
1
Chapter 1.
Random outcomes and random variables
5
1.
Dierential Equations (part 3): Systems of First-Order Dierential Equations
(by Evan Dummit, 2016, v. 2.00)
Contents
6 Systems of First-Order Linear Dierential Equations
1
6.1
General Theory of (First-Order) Linear Systems
. . . . . . . . . . . . . . . . .