Introduction to Linear Algebra and Differential Equations
MATH 24

Spring 2014
MATH 24: Linear Algebra & Differential Eqs  Discussion Section 8
Spring 2016
Vector spaces
1. Write down the definition of a vector space.
2. Prove that the following are vector spaces. (Note: You do not need to prove results known
from basic algebra but
Introduction to Linear Algebra and Differential Equations
MATH 24

Spring 2014
MATH 24: Linear Algebra & Differential Eqns  Discussion Section 9
Spring 2016
Secondorder linear homogeneous DEs with constant coefficients
1. Each of the following IVPs corresponds to a harmonic oscillator. Find the solution and specify its amplitude,
Introduction to Linear Algebra and Differential Equations
MATH 24

Spring 2014
Math 24 Final Exam: 123pm, May 12, 2006
Instructor: Boaz Ilan
READ ALL THE INSTRUCTIONS!
1. Write your name on the front of your bluebook as well as in the space below.
YOUR NAME:
2. This exam is closedbook and no calculators are allowed. You are allowed
Introduction to Linear Algebra and Differential Equations
MATH 24

Spring 2014
COMPLEX NUMBERS
Introduction:
A real number a can be graphically displayed as a point on a real number line
1
0
1
a
Imaginary axis
Similarly, we can display a complex number a + i b as a point in a complex plane
a+ib
b
r
a
Real axis
Instead of using Cart
Introduction to Linear Algebra and Differential Equations
MATH 24

Spring 2014
MATH 24: Linear Algebra & Differential Eqns  Discussion Section 2
Spring 2016
Direction fields
a) Match each of the following differential equations with one of the direction fields.
(i)
dy
= y 2 t2
dt
(ii)
dy
= t sin(y)
dt
(iii)
dy
= y2 9
dt
(iv)
dy
= y
Introduction to Linear Algebra and Differential Equations
MATH 24

Spring 2014
MATH 24: Linear Algebra & Differential Eqns  Discussion Section 1
Spring 2016
Differentiation & Integration Review
(1) Calculate the first nonzero term in a Taylor Series expansion for x3 about x = 0. What can you
say about other nonzero terms in the exp
Introduction to Linear Algebra and Differential Equations
MATH 24

Spring 2014
Math 24,
Midterm Exam 2 Solutions,
March 29, 2016
ON THE FRONT OF YOUR GREEN/BLUEBOOK WRITE: YOUR NAME AND A FOURPROBLEM
GRADING GRID. Show ALL of your work and box in your final answers. Unless otherwise mentioned, an answer without the relevant work wi
Introduction to Linear Algebra and Differential Equations
MATH 24

Spring 2014
MATH 24: Linear Algebra & Differential Eqns  Discussion Section 6
Spring 2016
Nonlinear Models
1. Draw the phase line (not the full graph) for the following equation, and use it to determine
the behavior of the system:
dy
= y + y2
dt
2. For the following
Introduction to Linear Algebra and Differential Equations
MATH 24

Spring 2014
MATH 24: Linear Algebra & Differential Eqns  Discussion Section 3
Spring 2016
Picards Theorem
(1) What can you conclude from Picards theorem?
(i)
dy
= t cos1 y,
dt
(ii)
dy
t
= ,
dt
y
y(0) = 1
y(0) = y0
(iii)
dy
1
= 2
,
dt
t + y2
(iv)
dy
= y 2/3 ,
dt
y(0)
Introduction to Linear Algebra and Differential Equations
MATH 24

Spring 2014
MATH 24: Linear Algebra & Differential Equations  Schedule
Week
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Date
Jan 19
Jan 21
Jan 26
Jan 28
Feb 2
Feb 4
Feb 9
Feb 11
Feb 16
Feb 18
Feb 23
Feb 25
Mar 1
Mar 3
Mar 8
Mar 10
Mar 15
Mar 17
Mar 2124
Mar 29
Mar 31
Apr
Introduction to Linear Algebra and Differential Equations
MATH 24

Spring 2014
MATH 24: Linear Algebra & Differential Eqns  Discussion Section 4
Firstorder linear DEs
1. Consider the following DE for y(x):
dy
1
= y +
dx
1 + ex
(a) Classify the equation.
(b) What can you say about solutions to the IVP with y(t0 ) = y0 ?
(c) Find th
Introduction to Linear Algebra and Differential Equations
MATH 24

Spring 2014
MATH 24: Extra Worksheet
Spring 2016
6.2 & 6.3 Linear Systems with Real & Complex Eigenvalues
1. Calculate the eigenvalues and associated eigenvectors for the matrix
2 4
A=
1 1
and determine the dimension of each eigenspace. Sketch the eigenvectors. Deter
Introduction to Linear Algebra and Differential Equations
MATH 24

Spring 2014
MATH 24: Linear Algebra & Differential Eqns  Discussion Section 7
Spring 2016
Conceptual review questions
1. What is a solution of a firstorder differential equation? How do you verify such a solution?
2. What is a solution of a firstorder initial valu
Introduction to Linear Algebra and Differential Equations
MATH 24

Spring 2014
J.G. Pieters, G.G.J. Neukermans, M.B.A. Colanbeen, Farmscale Membrane Filtration
of Sow Slurry, Journal of Agricultural Engineering, No. 73, 403409, 1999.
One way to assess the uses of membrane filtration is to test it against the most complex and
chall
Introduction to Linear Algebra and Differential Equations
MATH 24

Spring 2014
Lecture 13 : Sections 3.3 and 3.4
Spring 2017
March 2, 2017
Math 24 Spring 2017
Lecture 13
Review
Augmented matrices: The system of linear equations
1x1
2x1
1x1
+
+
1x2
3x2
2x2
+
+
1x3
1x3
2x3
=
=
=
3
8
3
is equivalent to the augmented matrix
1
1
1
3
2 3
Introduction to Linear Algebra and Differential Equations
MATH 24

Spring 2014
Lecture 11: Sections 3.2
Spring 2017
February 23, 2017
Math 24 Spring 2017
Lecture 11
Section 3.2: Systems of Linear Equations
Dimensions: An m n system of linear equations is of the
form:
A ~x = ~b
mn n1
m1
m is the number of equations
m is also the numb
Introduction to Linear Algebra and Differential Equations
MATH 24

Spring 2014
Lecture 9: Sections 3.1 and 3.2
Spring 2017
February 14, 2017
Math 24 Spring 2017
Lecture 9
Section 2.6: Systems of Differential Equations
PredatorPrey Example: It is known that foxes eat rabbits.
R(t)
population of rabbits
F (t)
population of foxes
How
Introduction to Linear Algebra and Differential Equations
MATH 24

Spring 2014
Lecture 12: Sections 3.2 and 3.3
Spring 2017
February 28, 2017
Math 24 Spring 2017
Lecture 12
Section 3.2: Systems of Linear Equations
Reduced Row Echelon Form (RREF)
An augmented matrix is in reduced row echelon form if
1
The leftmost nonzero entry of ea
Introduction to Linear Algebra and Differential Equations
MATH 24

Spring 2014
Lecture 10: Sections 3.2
Spring 2017
February 16, 2017
Math 24 Spring 2017
Lecture 10
Todays Lecture
We will
1
Interpret linear equations geometrically
2
Learn what elementary row operations are
3
Solve systems of linear equations using elementary row
ope
Introduction to Linear Algebra and Differential Equations
MATH 24

Spring 2014
Lecture 15 : Sections 3.5 and 3.6
Spring 2017
March 9, 2017
Math 24 Spring 2017
Lecture 15
Section 3.5: Vector Spaces
A set is a collection of objects.
Examples:
1
R3 = set of column vectors with 3 coordinates, each in R
x
= y : x, y , z R
z
2
3
R22 =
Introduction to Linear Algebra and Differential Equations
MATH 24

Spring 2014
Writing 116: Writing in the Natural Sciences
Habecker/ Spring 2017
Technical Explanation for a Lay Audience (TELA)
Topic:
Select a relatively current scientific topic that interests you; topics can be found in original research
articles, in the news, in p
Introduction to Linear Algebra and Differential Equations
MATH 24

Spring 2014
MATH 24: Linear Algebra & Differential Eqns  Discussion Section 5
Spring 2016
Growth & Decay
1. A 25 year old individual gets a job and begins earning $50,000 per year but has expenses of
$40,000 a year. She puts all her remaining money after expenses in
Introduction to Linear Algebra and Differential Equations
MATH 24

Spring 2014
WM
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