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 subtle task is to keep it simple and succinct. Your sol
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 wine from glass 1 and add it to glass 2. The wine and wa
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 = 1 1
1 0
1
1
1
(b)
1
B = 0
0
2. If A =
4
1
1
1
0
0
0
2
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:
0
0 1
1 0
and det 0
0
0 0
1
0 0
1
0
0
0
produce an up
Math 21242
Matrix Theory
Fall 2014
Problem Set 3
Due date: Next Wednesday
1. If a1 , ., an are vectors (incl. row and columnvectors) and x1 , ., xn are scalars then the vector k=1.n xk ak is called the linear combination of the vectors a1 , ., an with co
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 to the basis B
B := cfw_e1 , e2 , . . . , ek1 , ek + e
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 as its nullspace, and what matrix has P as its row space
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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 polynomial is factored into
det(A I) = (1 )(2 ) . . . (n ) ,
a
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 new basis B = cfw_b1 , ., bn is an arbitrary one. Cons
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 exist B, C such that AB = CA = I.
Lets call A Invertible (n
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 subtle task is to keep it simple and succinct. Your sol
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 statement below, write a sentence that expresses its negatio
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
(b) Manipulating logical expressions (Hint : Look at Ex
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 subtle task is to keep it simple and succinct. Your sol
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 numbers, S , where
S = cfw_3k + 5j  k, j N cfw_0
Solut
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 subtle task is to keep it simple and succinct. Your solut
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 g : R3 R2 by g (x, y, z ) = (xz, yz ). We rst prove tha
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 insucient.
Use the notation from the lectures, such as
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 counterexample if false:
(a) A is invertible.
(b) A is diagon
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 A, and a vector z orthogonal to the null space of A.
2
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)
x = (x1 , ., x4 ) R4  x2 = 3
(b)
x R4  x 2 = 0
(c)
x