MAS 3105
Quiz 4 and Key
AM: Feb 28, 2008
Prof. S. Hudson
1) [40 pts] Set B = cfw_v1 , v2 , v3 = cfw_(1, 0, 0)T , (1, 1, 0)T , (1, 0, 1)T . So, B is a set of 3
column vectors in R3 . Answer, and explain briey:
a) Is B a basis of R3 ?
b) Find the transitio
MAS 3105
Quiz 2
Jan 31, 2013
Prof. S. Hudson
1) [30 points] Find the determinants of these matrices. Even if you can do the work in your
head, include a remark about your reasoning, or some work. Label your answers clearly
(det A = . . .).
4441
17 2 300
4
MAS 3105
Quiz I and Key
Jan 17, 2013
Prof. S. Hudson
1) Solve this Leontief problem. The matrix A shows the ow of goods in a tribe with three
clans D, F and H in that order. For example, clan D gives 0.5 units of their production
to clan F each day. Assum
MAS 3105
Quiz I Key
Jan 19, 2012
Prof. S. Hudson
1) Use Gaussian elimination to put the following system into reduced row echelon form.
Use matrix notation. Find the solution set, using notation (if necessary) in your answer.
x2 + x3 =1
x1 + 2x2 + x3 =4
2
MAS 3105
Final Exam and Key
April 25, 2013
Prof. S. Hudson
1) [10pts] Suppose A is a 6 7 matrix and its column space is spanned by a1 and a3 ,
which are linearly independent (LI). Find the dimension of N (A) and explain briey.
2) [15 points] Choose ONE of
MAS 3105
Quiz III
Feb 16, 2012
Prof. S. Hudson
1) [40pt] Find a spanning set for N (A). Write your answer in set notation (as usual).
A=
0
1
31
11
2) [30pt] Find the (2,3) entry of B 1 from the
from the section on Cramers Rule).
1
B = 0
1
1
3
ratio of 2 d
MAS 3105
Quiz III and Key
Feb 14, 2013
Prof. S. Hudson
1) [20pt] This is a slight rephrasing of HW 3.1.11. Dene + on the set of column vectors
x1
x1
in R2 as usual, but dene scalar multiplication by
=
. Is R2 a vector
x2
x2
space with these operations ?
MAS 3105
Quiz 5 and Key
Mar 21, 2013
Prof. S. Hudson
1) [40pts] True-False:
The 3 3 identity matrix I is rank decient.
If A B then AT B T .
If A B then N (A) = N (B ).
If A B then rank A = rank B .
With the notation of Ch 5.1 for projections, | = |p|.
11
MAS 3105
Quiz 6 and Key
April 5, 2012
Prof. S. Hudson
1) Let S be the subspace of R3 spanned by x = (1, 0, 1)T and y = (0, 1, 1)T . Find an
orthonormal basis of S .
2) In C [0, ] with inner product
0
f g dx, compute 2, 1 + 3 cos(x) .
3) Answer True or Fal
MAS 3105
Quiz 5 and Key
March 22, 2012
Prof. S. Hudson
1) [30pts] Use the information below to compute the vector x100 R2 from the Rabbit
example, done in class. Recall that x0 = [1, 0]T and xn = Axn1 . The eigenvectors of A
were v1 = [1, 1]T and v2 = [1,
MAS 3105
Quiz 4 and Key
Feb 28, 2013
Prof. S. Hudson
1) Let u1 = (1, 1, 1)T and u2 = (1, 2, 2)T and u3 = (2, 3, 4)T .
a) Write x = (3, 2, 5)T as a linear combination of the uj . For maximal credit use a
transition matrix.
b) Write y = (1, 1, 2)T as a line
MAS 3105
Quiz 4 and Key
March 1, 2012
Prof. S. Hudson
1) Let v1 = (1, 2)T and v2 = (2, 3)T form a basis B for R2 . Find a pair of vectors w1 and
w2 so that S is the transition matrix from w1 and w2 to B .
S=
12
34
2) Answer True or False:
If the columns o
MAS 3105
Quiz 6 and Key
April 4, 2013
Prof. S. Hudson
1) [30 pts] Let S be the subspace of R4 spanned by x = (1, 2, 3, 4)T and y = (0, 1, 0, 1)T .
Find a basis of S . For a little EC [10 pts max], nd an orthonormal basis of S .
2) [40 pts] a) Use the norm
MAS 3105
Quiz 2 and Key
Feb 2, 2012
Prof. S. Hudson
1) [10 points] One MHW problem uses a command; oor(10*rand(6). Describe briey
what this means (what does MATLAB do with this? does it compute an integral ? a
column vector ? what would change if you ente
MAS 3105
Quiz I and Key
Jan 16, 2014
Prof. S. Hudson
1) Find the solution set, using notation, if necessary.
x1 + 3x2 + x3 =2
x2 x3 =4
2a) Given the info below, is the system Ax = c1 (where c1 = [11, 3]T )consistent ? Explain.
A=
1
1
0 4
1 0
,
1 4
B = 2 0
MAS 3105
Quiz 2, Key
Jan 31, 2014
Prof. S. Hudson
For problems 1 and 2, let
1
0
A=
0
0
1
1
0
2
0
1
1
2
1
2
1
5
1) Find an elementary matrix E so that EA is in RREF.
2) Compute the cofactor A21 .
3) Answer True or False. You do not have to justify your ans
MAS 3105
Quiz 2 and Key
Feb 2, 2012
Prof. S. Hudson
1) [10 points] One MHW problem uses a command; oor(10*rand(6). Describe briey
what this means (what does MATLAB do with this? does it compute an integral ? a
column vector ? what would change if you ente
NAME:
MT-12/MAS3105
Page 1 of 2
1. (2 pts.)
Compute the orthogonal projection of [1
the line through [-4
2]T and the origin.
7]T on
2. (4 pts.)
Let u1 = [3 1]T, u2 = [-2 6]T, and x = [-6
(a) Verify that cfw_ u1, u2 is an orthogonal basis for 2.
(b)
3]T.
MAS 3105
Quiz I
May 12, 2005
Prof. S. Hudson
NAME
Show all your work and reasoning for maximum credit. If you continue your work
on another page, be sure to leave a note. Do not use a calculator, book, or any personal
paper. You may ask about any ambiguou
MAS 3105
Quiz I and Key
Sept 13, 2006
Prof. S. Hudson
1
0
A=
0
3
0
0
5
1
0
0
1
1
1
0
0
1a) Assume A (above) is the augmented matrix for a linear system. List the lead variables
for the system.
1b) Find the solution set of the system.
1c) Use GE to put the
NAME:
Final Exam/MAS3105
Student Number:
Page 1 of 7
Exam Number:
Carefully use complete sentences and appropriate mathematical
notation in answering each of the following questions. Show all
essential work; observe that the line of reasoning communicated
NAME:
MidTest/MAS3105
Page 1 of 5
1. (36 pts.)
Using complete sentences and appropriate notation,
define each of the terms or items below. Give the most general
definition you have available at this time.
(a)
Linear combination :
(b)
Spancfw_v1,.,vm :
(c)
NAME:
MT-01/MAS3105
Page 1 of 2
1. (4 pts.)
Write down the augmented and coefficient matrices
for the following system of linear equations. Label each
appropriately, so they can be distinguished.
3x2 - 7x4 + x5 = 0
= 20
-2x1 - 4x3
3x4 - 11x5
= -6
2. (6 pt
NAME:
MT-02/MAS3105
1. (6 pts.)
Page 1 of 2
3x2 - 7x4 + x5 = 0
-2x1 - 4x3
3x4 - 11x5
= 20
= -6
(a) Write the system of linear equations above as an equivalent
vector equation.
(b)
Write the system of linear equations above in the form Ax=b.
(c) In what se
NAME:
1. (4 pts.)
through
MT-03/MAS3105
Page 1 of 2
(a) Find a parametric equation for the line
a = -3 and parallel to b = 23
5
-5 .
(b) Find a parametric equation for the line through a and b,
where
a = -3 and
5
b = 23
-5.
2. (6 pts.)
Using complete s
NAME:
MT-04/MAS3105
Page 1 of 2
1. (6 pts.)
Using complete sentences and appropriate notation,
define each of the items below.
(a)
Linear Transformation
(b)
Onto
(c)
One-to-one
2. (2 pts.)
Create two 2 x 2 matrices A and B with integer
entries so that AB
NAME:
MT-05/MAS3105
Page 1 of 2
1. (6 pts.)
Using complete sentences and appropriate notation,
define each of the terms below.
(a)
Invertible, regarding matrices
(b)
Invertible, regarding linear transformations
(c)
LU Factorization
2. (2 pts.)
Suppose the
NAME:
MT-06/MAS3105
Page 1 of 2
1. (5 pts.)
Suppose that A and B are 4 x 4 matrices with
det(A) = -2 and det(b) = 3. Using appropriate properties of the
determinant, compute each of the following.
(a)
det(AB)
=
(b)
det(A-1)
=
(c)
det(BT)
=
(d)
det(A5)
=
(
NAME:
1. (6 pts.)
MT-07/MAS3105
Page 1 of 2
Define each of the following terms.
(a)
Spancfw_v1,.,vm
(b)
Linearly independent
(c)
Basis
2. (4 pts.)
let
Let V be the first octant in xyz-space; that is,
x
V = y : x 0, y 0, z 0
z
.
Let "+" be the usual vect
NAME:
MT-08/MAS3105
Page 1 of 2
1. (4 pts.)
Using complete sentences and appropriate notation,
define each of the following terms.
(a)
Dimension
(b)
Rank
2. (6 pts.)
vector
(a) Find the vector x determined by the coordinate
[x]B =
when
x
B = cfw_ b1 ,b2 ,
Answers:
1) Start with BB 1 = I, but focusing only on column 3 of both sides, we get Bx = e3 .
B2
Then x2 = det
det B = 3/4.
But most people didnt use Cramers Rule, and used B 1 = det1 B adj B. The (2,3)
entry of adj B is -3, which leads to the same answe
entries! There were many correct answers, but the average grade on this problem was low,
perhaps because it was hard to get partial credit.
2) The answer should be written v = 3u w, or something very close to this. You can
calculate the coefficients x1 =
MAS 3105
Quiz III Key
Sept 29, 2016
Prof. S. Hudson
1) [20 pt] Use Cramers Rule to find the (2,3) entry of B 1 . Or, for partial credit, find the
answer using some other method from Ch.2.3.
2 1
4 3
2 2
1
B = 0
1
2) [15 pt] Find a spanning set for N (A). W
MAS 3105
Quiz I Key
Sept 2, 2016
Prof. S. Hudson
1) Use Gaussian elimination to put the following system into reduced row echelon form.
Use matrix notation. You dont have to find the solution set. For a little extra credit, find
the least number of GE ste
explain these. I did not require detailed proofs, but did require some fairly convincing
logic and/or calculations.
1a) Yes. S = N ([1 0 4]), so it must be a subspace.
1b) No. [3, 1, 1]T S but [6, 2, 2]T 6 S. So, this is not closed under .
1c) Yes. S = sp