F16 Study Guide for Exam 1
Linear Algebra, Dave Bayer
1
Our first exam will be held in class on Tuesday, October 11, 2016. Makeup exams will only be given under
exceptional circumstances, and require a note from a doctor or a dean.
Exam 1 will consist of
GU 4260 Market Design
Spring 17
Assignment ] 2
Question 1 Take any marriage problem in which agents on one side, say, women, have the
same preferences over the men. Is there a unique stable matching? If yes, provide a sketch
of the proof. If not, provide
6
F16 Study Guide for Exam 1
Linear Algebra, Dave Bayer
[7] By least squares, find the equation of the form y = ax + b that best fits the data
x1
x2
x3
x4
0
y1
1
y2
=
2
y3
3
y4
1
1
1
2
[8] Find an orthogonal basis for the subspace V of R4 spanned by
4
F16 Study Guide for Exam 1
Linear Algebra, Dave Bayer
[6] Find the intersection of the following two affine subspaces of R4 .
1
0
1
w
x
1
r
= 1 + 2
1 2 s
1
y
0
1
1
z
2
1
w
3
1
x
= +
0
1
y
1
1
z
1
1
t
1 u
1
(F15 Exam 1) (Solutions)
[1
8
F16 Study Guide for Exam 1
Linear Algebra, Dave Bayer
[3] Express A as a product of elementary matrices, where
3 5
A =
1 0
[4] Find the matrix A such that
1
0 1
3
0 0
1 1 = 3
3 0
A 1
0 1 1
0 3 0
[5] Find the intersection of the following two affine subs
3
F16 Study Guide for Exam 1
Linear Algebra, Dave Bayer
(F15 Homework 1) (Solutions)
[1] Solve the following system of equations.
w
1 1 1 3
3
1 1 2 3 x = 4
y
1 1 3 3
5
z
[2] Using matrix multiplication, count the number of paths of length nine from
10
F16 Study Guide for Exam 1
Linear Algebra, Dave Bayer
[2] Using matrix multiplication, count the number of paths of length eight from x to z.
x
y
z
[3] Express A as a product of elementary matrices, where
0 1 0
A = 2 2 0
0 0 1
[4] Find the matrix A suc
5
F16 Study Guide for Exam 1
Linear Algebra, Dave Bayer
[5] Find the intersection of the following two affine subspaces of R4 .
1
0
0
2
w
r
x
1
0
s
= 0 + 1
0 1
0
y
1
t
0
0 1
0
z
1
0
w
x
= 1 + 2 u
1
2
y
0
3
z
(F15 Homework 2)
[1] Let v1
9
F16 Study Guide for Exam 1
Linear Algebra, Dave Bayer
[3] Express A as a product of elementary matrices, where
0 1 12
A = 1 0 0
0 0 3
[4] Find the matrix A such that
1
0 1
0 0 3
1 1 = 0 0 3
A 1
0 1 1
0 0 3
[5] Find the intersection of the following two
7
F16 Study Guide for Exam 1
Linear Algebra, Dave Bayer
[5] Let V be the vector space R3 , equipped with the inner product
d
2 1 0
1 2 1
e
h(a, b, c), (d, e, f)i = a b c
0 1 2
f
Using this inner product rather than the dot product, find the orthogonal
F16 Study Guide for Exam 1
Linear Algebra, Dave Bayer
2
Homework will count as 10% of your course grade. The homework for Exam 1 is given below. You may turn in
problems in batchs as you complete them; homework that is received early will receive more car