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
Answers: 1)
1 2 1
0 0 2
 7
 8
1 2 1
0 0 1
 7
 4
1
0
2 0
0 1


3
4
So, it can be done in 2 steps. Not less, because changes are required in both row 1 and
in row 2 (because the RREF is unique). One step cannot change two rows, unless it is a
swap, wh
MAS 3105
Quiz 2
Sept 16, 2016
Prof. S. Hudson
1) Find a 3x3 matrix A with at least two nonzero entries such that A2 = O (the zero
matrix). If this is not possible, explain why not.
2) Write v as a linear combination of u and w. For maximum credit, solve t
MAS 3105
Quiz 4 Key
Oct 14, 2016
Prof. S. Hudson
1) Which of the following are subspaces of R3 ? Answer Y/N four times and explain each
answer briefly, perhaps by writing S another way.
a) S = cfw_(x1 , x2 , x3 )T  x1 + 4x3 = 0
b) S = cfw_(x1 , x2 , x3 )
Name: _
MAT 261Exam 2
Due: 05/31/16
Directions: Please read each question carefully. Be sure to show your working in order to get the full credit you
deserve.
1. Let
[ ] []
=
1
5
2
12
, 1 =
1
3
, 2 =
[]
1
5
, and 3 =
[]
2
6
Solve the following equations
Syllabus Mathematics 261  Linear Algebra
Text: Linear Algebra with Applications, 9th Edition, Steven J. Leon
Goals: The goal of the course is the mastery of the following core topics as well as
increasing problemsolving capability.
Matrix Addition and
MAS 3305 — LINEAR ALGEBRA
TEST #2  SPRING 2014
FLORIDA INT'L UNIV.
TIME: 75 min.
Answer all 6 questions. NO CALCULATORS or CELL PHONES ARE ALLOWED.
Show all working and provide all reasoning where required. An unjustiﬁed answer
will receive little or no
MAS 3105  LINEAR ALGEBRA . FLORIDA INT'L UNIV.
TEST #1 — SPRING 2014 . TIME: '75 min.
Answer all 6 questions. NO CALCULATORS or CELL PHONES ARE.ALLOWED.
ShOW’all working in problems 1—4. .Provide all reasoning in problems
5—6. An unjustified answer will
MAS 3105  LINEAR ALGEBRA I FLORIDA INT'L UNIV.
TEST #1  SPRING 2016 L TIME: 75 min.
Answer all 6 questions. N0 CALCULATORS or CELL PHONES ARE ALLOWED. Show
all working in problems 14. _ Provide all reasoning in problems 5 6. An unjustified answer
will
MAS 3105  LINEAR ALGEBRA I FLORIDA INT'L UNIV.
TEST #1  SPRING 2016 L TIME: 75 min.
Answer all 6 questions. N0 CALCULATORS or CELL PHONES ARE ALLOWED. Show
all working in problems 14. _ Provide all reasoning in problems 5 6. An unjustified answer
will
MAS 3105
Final Exam and Key
April 23, 2014
Prof. S. Hudson
1) Let S = span cfw_(1, 2, 3, 0)T , (1, 0, 0, 1)T , a subspace of R4 . Find a basis of S . For
about 5 points extra credit, nd an ONL basis of it.
2) [15 pts] Suppose a Markov process has the tran
MAS 3105
Quiz 5 and Key
March 21, 2014
Prof. S. Hudson
1) These two matrices are row equivalent. U is in REF but not RREF.
5
3
A=
2
0
2
0
5
5
2
0
6
3
1 0
0 1
U=
0 0
1
1
0
7
3
1
a) Find a dependency relation for the columns of A; give a nontrivial LC equal
MAS 3105
Quiz 6 and Key
April 4, 2014
Prof. S. Hudson
1) Find the best Least Squares t by a linear function for this data. One of the incomplete
GE calculations below might help a little.
x
1
0
1
2
y
0
1
3
9
4
2
2
13
21
1
0
2) In P3 with inner product
1/
MAS 3105
Quiz 3
Feb 14, 2014
Prof. S. Hudson
1) Answer all three parts, based on the matrix A given below.
a) Find det A and det (A1 ).
b) Find adj A.
c) Use the answer to b) to nd A1 .
3 1
A = 2 0
0 0
5
4
3
2) Which of the following are subspaces of R3 ?
MAS 3105
Quiz 4 and Key
Feb 28, 2014
Prof. S. Hudson
1) Let S be the subspace of P4 consisting of all polynomials of the form ax2 + bx + 4b.
Find a basis for S.
2) Find all values of k such that [k + 1, k + 2, k + 3]T is in the column space of A:
1
A = 1
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 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 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