MATH 2940 Academic Excellence Workshop Week 13
Parmeet Singh (ps732) & Alyse SanFilippo (as2578)
1. Circle each as either True or False.
(a) If A is a symmetric matrix, then A is invertible.
False
True or
(b) Any real symmetric matrix is diagonalizable.
o

MATH 2940 Academic Excellence Workshop Week 2
Parmeet Singh (ps732) & Alyse SanFilippo (as2578)
1. Decide whether each subset of is linearly independent or dependent
(a)
(b)
(c)
(d)
2. Suppose that and are vectors in .
(a) Show that if is a linearly indep

MATH 2940 Academic Excellence Workshop Week 3
Parmeet Singh (ps732) & Alyse SanFilippo (as2578)
1. Write a matrix equation that determines the loop currents. (Feel free to solve the system
later if you finish the worksheet early)
2. Determine by inspectio

MATH 2940 Week 3 Solutions
1.
2. a) Compare the corresponding entries of the 2 vectors. The second seems to be -3/2 times the
first vector. However, this doesnt hold for the third pair of entries. Thus, neither of the vectors is
a multiple of the other, a

Math 2940 AEW Week 2 Solutions
(1) (a) There is no quick way to just look at the set of vectors and determine whether or not it is
linearly independent in this case. A sure-fire algorithmic approach is just to write the vector
equation as a matrix equatio

Math 2940 AEW Week 1 Solutions
(1) When the second equation is replaced by its sum with 3 times the first equation, the system
becomes
2x1 x2 = h
0 = k + 3h
If k + 3h is nonzero, the system has no solution. Therefore, the system is consistent for any
valu

Math 2940 AEW Week 1 Solutions
(1) When the second equation is replaced by its sum with 3 times the first equation, the system
becomes
2x1 x2 = h
0 = k + 3h
If k + 3h is nonzero, the system has no solution. Therefore, the system is consistent for any
valu

Math 2940, Prelim 1
September 29, 2011
You are NOT allowed calculators or the text. SHOW ALL WORK!
2
3
1 0
2
1 5 by the following row operations in the
1) A matrix A was reduced to A0 = 4 0 1
0 0
0
given order:
2 times Row 1 subtracted from Row 2, 4 times

July 8, 2013
Prelim 1
Math 2940
sh is
ar stu
ed d
vi y re
aC s
o
ou urc
rs e
eH w
er as
o.
co
m
Please, except on the True/False question number 6 show your
reasoning and all your work. This is a 75 minute exam. Calculators are
not needed or permitted. Go

Discrete Dynamical Systems
I
x0 , x1 , x2 , . . . vectors in Rn .
I
x0 , x1 , x2 , . . . vectors in Rn .
I
xk represents the state of a dynamical system at time
k = 0, 1, 2, . . .
I
x0 , x1 , x2 , . . . vectors in Rn .
I
xk represents the state of a dynam

Applications to Linear Models
Linear Models in Statistics
I
The task: understand the connection between several
observed quantities in presence of uncertainty.
Linear Models in Statistics
I
The task: understand the connection between several
observed quan

Difference Equations
Space S of signals
I
A discrete-time signal is a sequence of numbers cfw_yk :
(. . . y1 , y0 , y1 , y2 , . . .).
Space S of signals
I
A discrete-time signal is a sequence of numbers cfw_yk :
(. . . y1 , y0 , y1 , y2 , . . .).
I
A sign

Least Squares Problems
Approximate solutions to linear systems
I
Suppose we want to solve for x a linear system Ax = b.
Approximate solutions to linear systems
I
Suppose we want to solve for x a linear system Ax = b.
I
If b does not belong to Col A, the s

Inner Product, Length and Orthogonality
I
u, v vectors in Rn .
I
u, v vectors in Rn .
I
We can view u and v as n 1 matrices.
v1
u1
v2
u2
u = . , v = . .
.
.
.
un
vn
I
u, v vectors in Rn .
I
We can view u and v as n 1 matrices.
v1
u1
v2

Orthogonal Sets
I
A set of vectors cfw_u1 , . . . , up in Rn is an orthogonal set
if any two distinct vectors in the set are orthogonal.
I
A set of vectors cfw_u1 , . . . , up in Rn is an orthogonal set
if any two distinct vectors in the set are orthogo

Eigenvectors and Linear Transformations
The matrix of a linear transformation
I
V , W vector spaces.
The matrix of a linear transformation
I
V , W vector spaces.
I
V n-dimensional, W m-dimensional.
The matrix of a linear transformation
I
V , W vector spac

The Gramm-Schmidt Process
I
cfw_x1 , . . . , xp a linearly independent set in Rn .
I
cfw_x1 , . . . , xp a linearly independent set in Rn .
I
Let W = Span(x1 , . . . , xp ).
I
cfw_x1 , . . . , xp a linearly independent set in Rn .
I
Let W = Span(x1 , .

Orthogonal Projections
I
y Rn , W a subspace of Rn .
I
y Rn , W a subspace of Rn .
I
The goal: find the point y in W closest to y.
I
y Rn , W a subspace of Rn .
I
The goal: find the point y in W closest to y.
I
The approach: find the point y in W such tha

MATH 2940 AEW Week 4 Solutions
(1)
(a) D = [ Compute:
C=, C=, C=
=,
=,
CD = C =
=
(b) Since D has 3 columns, and C has 2 rows, the product DC is undefined. It does not have
corresponding columns and rows and therefore cannot be multiplied.
(c) Left-multip

MATH 2940 Academic Excellence Workshop Week 4
Parmeet Singh (ps732) & Alyse SanFilippo (as2578)
1. C =
D=
a)
Compute CD.
b)
Compute DC.
c)
Suppose P is an invertible matrix and A = PBP-1. Solve for B in terms of A. (Hint:
Consider the associative property

Week 6 Solutions
(1) Suppose that are perpendicular nonzero vectors in We have the linear relationship ,
multiplying v to both sides gives us: Since by assumption then we must have c = 0. A similar
approach by multiplying w to both sides of the equation s

MATH 2940 Academic Excellence Workshop Week 12
Parmeet Singh (ps732) & Alyse SanFilippo (as2578)
1. Let = and = . Compute and compare (a)-(d) without using the Pythagorean Theorem.
a) ,
b)
c)
d)
e) Show that if x is in both W and W, then x = 0.
2. a) Let

AEW Week 13 Solutions
(1) (a) False. Consider the zero matrix as a counterexample.
(b) True. The result is guaranteed by the spectral theorem (page 397 in your book).
(c) True. Although not obvious one can show that the sum of two symmetric matrices, and

MATH 2940 Academic Excellence Workshop Week 11
Parmeet Singh (ps732) & Alyse SanFilippo (as2578)
1. A = , B =
a) Diagonalize A.
b) Compute B8.
c) Construct a nondiagonal 2x2 matrix that is diagonalizable but not invertible.
2. Let B = cfw_b1, b2, b3 be a

Week 10 Solutions
(1) a) The standard matrix A of T is
A = Col A =
b) A = Nul A =
c) A = basis for Col A =
d) basis for Nul A =
e) rank A = 3, dim Nul A = 1 - check that rank A + dim Nul A = n
(2) a) Choose a column or row to expand across (here I chose 4

MATH 2940 Academic Excellence Workshop Week 8
Parmeet Singh (ps732) & Alyse SanFilippo (as2578)
1. C
2. A
3. (a)
4. A mouse is put in the maze below. Each time period the doors in the maze are opened and it is
allowed to move to another room. 50% of the t

MATH 2940 Academic Excellence Workshop Week 9
Parmeet Singh (ps732) & Alyse SanFilippo (as2578)
1. Use Cramers Rule to solve the following system:
2. Use the adjugate to find the inverse of the following matrix
3. Show the following gives an equation of a

Week 8 AEW Solutions
(1) (a) P = , x0 =
(b) x4 =
(c) (P-I) = =
~
x2 = probability vector
Therefore 83.333% of days are sunny after a long period of time and 16.667% of days are rainy.
(2)
(a) FALSE The bottom row of P I is the negative of the sum of the o

MATH 2940 Academic Excellence Workshop Week 6
Parmeet Singh (ps732) & Alyse SanFilippo (as2578)
1. Prove that a set of two perpendicular nonzero vectors from is linearly independent when
[Orthogonal vectors are of great interest in chapter 6].
2. Find a b

PMATH 2940 PRELIM I
February 25, 2016
Name: _
Cornell NetID: _
_
Check the box corresponding to your discussion section:
W 7:30-8:20P
W 7:30-8:20P
R 9:05-9:55A
R 9:05-9:55A
R 10:10-11:00A
R 11:15-12:05P
R 12:20-1:10P
R 1:25-2:15P
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210 Liu
F 9