MATH 4063-5023
Homework Set 1
1. Let F be a eld, and let Fn denote the set of n-tuples of elements of F, with operations of scalar
multiplication and vector addition dened by
[
[
1; : : : ;
n]
1; : : : ;
n]
:
= [ a1 ; : : : ; a n ]
+[
1; : : : ;
n]
:
=[
1
MATH 4063-5023
Homework Set 5
1. Find the
20
(a) 0 2
00
characteristic polynomials and minimal polynomials of the following matrices.
0
0
2
The characteristic polynomial of this matrix is
2
0
0
2
0 = (2 )3
pA () = det 0
0
0
2
The minimal polynomial will
MATH 4063-5023
Solutions to Homework Set 6
1. Determine if the following matrices A are diagonalizable. When A is diagonalizable, provide both the
matrix C diagonalizing A and its diagonal form D (so that D = C1 AC).
(a) A =
2
0
1
1
We have
det (A I) = (
MATH 4063-5023
Solutions to Homework Set 4
1. Let P be the vector space of polynomials with indeterminant x.Which of the following mappings are
linear transformations from P to itself
(a) T : p xp
Let p1 , p2 be two polynomials and let , F. We have
p1
=
MATH 4063-5023
Solutions to Homework Set 3
1. Let F be a eld with exactly two elements (it will be isomorphic to Z2 ) and let V be a 2-dimensional
vector space over F. How many vectors are there in V ? How many dierent bases are there for V ?
Since any e
MATH 4063-5023
Homework Set 3
1. Let F be a eld with exactly two elements (it will be isomorphic to Z2 ) and let V be a 2-dimensional
vector space over F. How many vectors are there in V ? How many dierent bases are there for V ?
2. Test for the solvabili
MATH 4063-5023
Homework Set 4
1. Let P be the vector space of polynomials with indeterminant x.Which of the following mappings are
linear transformations from P to itself
(a) T : p xp
(b) T : p 2p
(c) T : p
dp
dx
(d) T : p
1
0
+ 2p
p (x) dx
2. Compute t
MATH 4063-5023
Homework Set 6
1. Determine if the following matrices A are diagonalizable. When A is diagonalizable, provide both the
matrix C diagonalizing A and its diagonal form D (so that D = C1 AC).
(a) A =
2
0
1
1
(b) A =
1
1
2
1
(c) A =
3
2
2
1
2.
MATH 4063-5023
Homework Set 5
1. Find the
2 0
(a) 0 2
0 0
2
(b) 0
0
characteristic polynomials and minimal polynomials of the following matrices.
0
0
2
1
1
2
1
2
0
2. Find the eigenvalues of the following matrice, and then for each eigenvalue, nd a basi
Math 4063-5023
SOLUTIONS TO FIRST EXAM
2:30 3:20 , October 23, 2013
1. Denitions. Write down the precise denitions of the following notions. (3 pts each)
(a) dependence relation.
A dependence relation amongst a set of vectors cfw_v1 , . . . , vk in a ve
MATH 4063-5023
Solutions to Homework Set 1
1. Let F be a eld, and let Fn denote the set of n-tuples of elements of F, with operations of scalar
multiplication and vector addition dened by
for all F and all [1 , . . . , n ] in Fn
[1 , . . . , n ]
:
= [a1
MAE 3323 Mechanical Design 1
Course Project
PROBLEM STATEMENT:
Consider the tower shown in the figure. Due to its good weld ability, we plan to use ASME SA
285 Grade C material for fabricating the pressure vessel with an inside diameter of 8ft. In the
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