21127: Concepts of Mathematics
Homework 1 (due Thursday, September 08)
Directions: Write up carefully argued solutions to the following problems. The rst task is to be complete and correct. The more
21127: Concepts of Mathematics
Homework 1 Solutions
1.22 (10 points)
We have two identical glasses. Glass 1 contains x ounces of wine; glass 2 contains x ounces of water
(x 1). We remove 1 ounce of w
Math 21242
Matrix Theory
Spring 2014
Problem Set 8
Due date: Next Wendesday
1. Find the eigenvalues and eigenvectors of the following matrices. Determine if they are diagonlaizable or not.
(a)
1 0
A
Math 21242
Matrix Theory
Spring 2014
Problem Set 7
Due date: next Wednesday
1. Count row exchanges to nd these
0 0
0 0
det
0 1
1 0
2. By applying row operations to
1
0
2
1
det
1 2
0
2
determinants:
Math 21242
Matrix Theory
Fall 2014
Problem Set 4
Due date: Next We
1. Suppose the
0
1
0
0
matrices in P A = LU are
0 0 1 3 2
1
1 0 0
0 0 0 2 1 4 2 1 0
=
0 0 1 4 2 9 1 4 1
2 1 5 1 5
2
0 1 0
0
1
1
1
Math 21242
Matrix Theory
Fall 2014
Problem Set 11
Due date: Next Friday
1. Let be an arbitrary scalar and 1 j < k n be xed. We consider a simple basis change
in Rn (or in Cn ) from the standard basis
Math 21242
Matrix Theory
Fall 2014
Problem Set 6
Due date: next Wednesday
1. Let P be the hyperplane in R3 with equation x + 2y z = 0. Find a vector perpendicular to
P . What matrix has the plane P a
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Math 21242
Matrix Theory
Spring 2014
Problem Set 10
Due date: Next Friday
1. Show that the determinant of a matrix A equals the product of its eigenvalues by imagining
that the characteristic polynom
Math 21242
Matrix Theory
Spring 2014
Problem Set 5
Due date: Next Wednesday
1. The most common case of basis change is when the original basis is the standard basis
E = cfw_e1 , ., en in Rn and the
Math 21242
Matrix Theory
Spring 2014
Problem Set 2
Due date: next Wednesday
1. Let A be an n n matrix. Recalling the denition, A is invertible i there exists a left and
right inverse, i.e. there exis
21127: Concepts of Mathematics
Homework 2 (due Thursday, September 15)
Directions: Write up carefully argued solutions to the following problems. The rst task is to be complete and correct. The more
21127: Concepts of Mathematics
Homework 2 Solutions
2.4 (10 points)
Let A and B be sets of real numbers, let f be a function from R to R, and let P be the set of positive
real numbers. For each state
21127: Concepts of Mathematics
Homework 3 Solutions
A7 (10 points)
Prove that the logical expression S is equivalent to the logical expression S (R R) in two
dierent ways by:
(a) Using a Truth Table
21127: Concepts of Mathematics
Homework 3 (due Thursday, September 22)
Directions: Write up carefully argued solutions to the following problems. The rst task is to be complete and correct. The more
21127: Concepts of Mathematics
Homework 4 Solutions
A9 (10 points)
Determine the set of natural numbers that can be expressed as the sum of some nonnegative number
of 3s and 5s. That is, the set of
21127: Concepts of Mathematics
Homework 4 (due Thursday, October 06)
Directions: Write up carefully argued solutions to the following problems. The rst task is to be complete and correct. The more su
21127: Concepts of Mathematics
Homework 5 Solutions
A11 (10 points)
Dene the function g : R3 R2 by g (x, y, z ) = (xz, yz ). Prove that g is surjective but not injective.
Solution:
Dene the function
Math 21242
Matrix Theory
Spring 2014
Problem Set 1
Due date: Wednesday Jan 29
In order to receive credit for your homework it is necessary to show all your intermediate work.
The solutions alone are
Math 21242
Matrix Theory
Spring 2014
Problem Set 9
Due: next We
1. If the eigenvalues of A (a 3 3 matrix) are 1, 1, 2, which of the following are certain to be
true? Give a reason if true or a counte
Math 21242
Matrix Theory
Spring 2014
Problem Set 6
Due date: next Wednesday
1. Let
1
A = 2
3
2
4
6
1
3
4
Find a vector x orthogonal to the row space of A, a vector y orthogonal to the column space
of
Math 21242
Matrix Theory
Spring 2014
Problem Set 3
Due date: Next Wednesday
1. Which of the following sets are subspaces? Carefully justify your answer. Describe the subspaces in geometric terms.
(a)