1. (25 pts) Given the following system of linear equations
3:1 + 2222 — x3 = 2
3121 + 4.722 + 2223 = —2
$1 + 4302 -— 61133 = 10
(a) (2 pts) Write the associated Augmented Matrix A.
(b) (9 pts) Use the row reduction algorithm to ﬁnd the Echelon Form of the
The first exam will be held in class on Friday September 14th. Please bring pencils
and a pen to the exam. You may bring a simple calculator to the exam to help with
basic arithmetic, but no scientific or graphing calculators will be allowed. Note that
yo
The second exam will be held in class on Friday October 12th. Please bring pencils
and a pen to the exam. You may bring a simple calculator to the exam to help with
basic arithmetic, but no scientific or graphing calculators will be allowed. Note that
you
The third exam will be held in class on Friday November 9th. Please bring pencils and
a pen to the exam. You may bring a simple calculator to the exam to help with basic
arithmetic, but no scientific or graphing calculators will be allowed. Note that you
Gage Carr
Math 1553
Prof. Srinivasan
11/28/16
Exercise One
Definitions:
Eigenvectors- a vector, that when row operations are performed, creates a vector that is a scalar
of itself
Eigenvale-the scalar of the eigenvector to make the orginial vector the eig
Georgia Institute of Technology
School of Mathematics
Math 1522 A Summer 2013
Midterm Exam 1 - sample
Friday 05-31-2013
Name (Print):
INSTRUCTIONS
This test has 4 problems on 6 pages (including the cover sheet and a page for
scrap work)
Look over your t
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5‘9wa ' Xu'Lzﬁ‘i
_ $3:—3
X
1. (25 pts) Given the following system of linear equations
3211 — $2 + 3:133 + 4x4 2 10
$1—$2+2:I:3+:E4 = 1
21134 + 3:123 — 21172 + 2:131 = 5
(a) Write the associated Augmented matrix A.
(b) Use the Bow Reduction Al
1. (13 pts) Given the Augmented Matrix
0 1 1 —1
3 7 —2 h
1 2 —1 —1
(a) (9 pts) Use the row reduction algorithm to ﬁnd the Echelon Form of the
Augmented Matrix. Show each step of the row reduction algorithm in a
new matrix.
(b) (4 pts) Determine the value(
Second quiz for Math1522 T
Sept. 12nd
1.
Note that
1
2
3
3
1
1
4
=
1
1
1
1
2
2
1
Use this fact (without row operations) to nd a solution of linear system
as follow:
x1 + 2x2 3x3 + x4 = 4
2x1 3x2 + x3 x4 = 1
2. We have a matrix
2
A= 3
8
(2 points)
eq
Math 1522, Exam 2: 2.1-3,5,8-9, 3.1-3, 4.1-3 (practice)
1. (25 points) (2.3.20,21) If A is a 3 3 matrix and
2
1
A 1 = A 0 ,
7
0
then show the equation Ax = 0 has more than one solution.
Solution: Notice that x = 0 (the zero vector) is
would be
2
1
1 0 =
Math 1522, Quiz 4: 2.8-3.2 (practice)
Name/Section:
1. (50 points) (2.8.9) Let
2 3 4
8
6 .
A = 8
6 7 7
Is
6
v = 10
11
in the null space of A?
Solution: Remember that the null space of A is cfw_x : Ax = 0. Thus, we simply
need to calculate Av and see if i
Math 1522, Exam 2: 2.1-3,5,8-9, 3.1-3, 4.1-3
1. (25 points) (2.3.20,21) If A is a 3 3 matrix and
1
4
2 = A 5 ,
A
3
6
then nd
10
A 10 .
10
Solution: Notice that
0
4
1
3
0 = A 5 A 2 = A 3
0
6
3
3
Therefore,
3
0
10
10 = 10 A 3 = 0 .
A
3
3
0
10
2. (25 poi
Math 1522, Quiz 2: 1.5-9 (practice)
Name/Section:
1. (50 points) (1.7.1) Is the collection of vectors linearly dependent or linearly independent?
7
9
5
0 , 2 , 4
0
6
8
Solution: The matrix with these vectors as columns and its reduction are as follows
Math 1522, Quiz 1: 1.1-5 (practice)
Name/Section:
1. (50 points) (1.1.13) Solve the system by Gaussian elimination.
x1
3x3 =
8
2x1 +2x2 +9x3 =
7
x2 +5x3 = 2
Solution: The augmented matrix associated with this system and its reduction are
as follows:
1 0 3
Math 1522, Exam 1: 1.1-5, 1.7-9 (practice)
Name/Section:
1. (25 points) (1.2.13) Find the general solution of the system whose augmented matrix is
1 3 0 1
0 2
0
10
0 4
1
0
00
1
9
4
0
00
0
0
0
Solution: This system is already in echelon form with
and fourt
Math 1522, Quiz 2: 1.5-9 (practice)
Name/Section:
1. (50 points) (1.7.5) Is the collection of vectors linearly dependent or linearly independent?
9
3
0
2 1 7
,
,
1 4 5
2
4
1
Solution: The matrix with these vectors as columns and its reduction are as fo
Math 1522, Quiz 3: 2.1-5 (practice)
Name/Section:
1. (50 points) (2.1.9) Let
A=
23
1 1
19
3 k
and
.
Find all values of k such that AB = BA.
Solution: The product matrices are
AB =
7 18 + 3k
4 9 + k
and
BA =
7
12
6 k 9 + k
.
In order for these to be the sa
Math 1522, Quiz 4: 2.8-3.2 (practice)
Name/Section:
1. (50 points) (2.8.10) Let
2 2
0
3 5 ,
A= 0
6
3
5
and let L be the linear transformation determined by L(x) = Ax. Is
5
v= 5
3
in the kernel of L?
Solution: Remember that the kernel is cfw_x : L(x) = 0.
Math 1522, Quiz 5: 4.4-7 (practice)
Name/Section:
1. (50 points) (4.4.5) If the vector
1
1
x=
,
is expressed in the basis
1
2
,
3
5
,
what are its new coordinates?
Solution: The new coordinates are
c1
c2
where
1
1
= c1
1
2
3
5
+ c2
.
These values may be f
Math 1522
Practice Test 5
Geronimo
1.a Find the best linear t to the points (xi , yi ) i = 1.3 given by,
(1, 2) (2, 3), (3, 2)
1.b Suppose that C is an m m matrix with linearly independent columns. Show that
C T C is inverible.
3
2.a Let u = 1 and W is th
Math 1522
Practice Test 3
Geronimo
1. Find the range and null space of the following matrices.
spaces.
1122
1
2 3 1 5
1
(a) A =
(b) A =
1213
1
Find a basis for each of these
2
1
1
3
1
1
2. Compute the determinant of the following matrices
(a)
2101
1 2 1
Math 1522 1. Consider
Practice Test 2 2 A = 1 0 0 1 1 1 1. 2
Geronimo
(a) If it exists find A-1 (b) Write out the elementary matrices you used to perform the first three elementary row operations in the above calculation. 2. Determine between which spaces
Math 1522 Practice Final Exam 1a. Find the eigenvalues and corresponding eigenvectors to the matrix A= 1 4 -2 -8
Fall 07
b. Find A15 (Show all work). c. Let A be an nn matrix. Show that eigenvectors associated with different eigenvalues of A are linearly