Math 317
Exam 1
Name:
Directions: Please answer all questions. Pen or pencil only. To receive full credit all work or necessary
justications must be given.
1. Find the reduced row echelon form and the rank
123
2 4 5
246
of the matrix
1
1 .
2
Solution:
1
2
Exercise 3.3.8
Suppose cfw_v, w is a linearly independent set. We claim that the set cfw_v
w, 2v + w is linearly independent as well. To see that this is the case we
consider the equation
x1 (v w) + x2 (2v + w) = 0
and show that only solution is the triv
Assignment #3: Rhetorical
Analysis of Text
Excellent
Good
(Writer responds thoughtfully and
creatively, requiring little or no revision)
(Writer responds fully, requiring
some revision)
Clear and engaging articulation of
context and purpose; the reader is
Exercise 2.2.7
a. Let
cos
sin
A =
sin
cos
,
A =
cos
sin
sin
cos
.
Then observe that
A A =
=
cos
sin
sin
cos
cos sin
sin cos
cos cos sin sin
sin cos + cos sin
(sin cos + cos sin )
cos cos sin sin
.
Moreover,
A A =
=
cos sin
sin cos
co
1
The Fibonacci Sequence
Exercise 1
Let
0
1
A=
1
1
,
1
1
x0 =
.
a. By straightforward matrix multiplication we get
A2 =
1
1
1
2
1
2
A3 =
,
2
3
2
3
A4 =
,
3
5
3
5
A5 =
,
5
8
.
b. Next we get the following from matrix-vector multiplication, using our comput
Exercise 1.1.16
Let x = AB and y = AD. To show that P lies on the diagonal AC it suces to show that AP is a scalar
multiple of x + y. First notice that
1
y + DE = x;
3
1
so DE = 3 x y. Next notice that
3 1
3
DP = DE = x y.
4
4
4
Therefore we have
1
AP
Exercise 1.4.11b
The set of all vectors x that are orthogonal to both (1, 1, 1, 1) and (1, 2, 1, 1), is precisely the solution
set of the following system of linear equations:
x1 + x2 + x3 x4 = 0
x1 + 2x2 x3 + x4 = 0.
To this end, we consider the augmente
Math 317 Exam 2 Name:
Directions: Please answer all questions, Pen or pencil only. To receive full credit all work or necessary
justifications must be given.
ll Consider the matrix
S , a , U ~ f U 1'" f
" é i E l I
y ~ « . > < r a l , A
(f; 7 i
Exercise 2.5.8
Suppose A is invertible. Then observe that part 4 of Proposition 5.1 in
Section 2.5
(A1 )T AT = (AA1 )T = I T = I.
Similarly,
AT (A1 )T = (A1 A)T = I T = I.
We conclude that (A1 )T is a left and right inverse of AT , which implies that
AT i
Exercise 4.1.2
Observe that if PV = A(A A)1 A , then
PV = A(A A)1 A
= A(A A)1 ) A = A(A A) )1 A
= A(A A)1 A = PV ,
where the second and fourth equalities follow from the formula (CD) = D C ,
while the third equality follows from the formula (C 1 ) = (C )1
Exercise 3.6.9
Let A, B be n n matrices with (i, j ) entries denoted aij and bij respectively.
a. Note that since the (i, j ) entry of A is equal to aji , it follows that the
diagonal entries of A are equal to those of A. Therefore tr A = tr A .
b. Note t
Exercise 3.2.10
Let A be an m n matrix and B ben an n p matrix. We remark that in
parts b. and d. below it will be important to note that for any m n matrix
A, C (A) = cfw_b Rm : Ax = b has a solution.
a. We rst claim that N (B ) N (AB ). Indeed, let v N
Math 317: Linear Algebra
Exam 1
Fall 2015
Name:
*No notes or electronic devices. You must show all your work to receive full credit. When
justifying your answers, use only those techniques that we learned in class up to Section
2.2. Good luck!
1. Let u =