18.700 Problem Set 1 Solutions
1. (3 points) Consider the set of complex numbers
G = cfw_a + bi | a, b Q.
(The G stands for Gauss; these numbers should be called Gaussian rational numbers, although I dont know if they actually are.) Is G a eld (with the s
Gaussian elimination
October 14, 2013
Contents
1 Introduction
1
2 Some denitions and examples
2
3 Elementary row operations
7
4 Gaussian elimination
11
5 Rank and row reduction
16
6 Some computational tricks
18
1
Introduction
The point of 18.700 is to und
18.700. Exam 1. Fall 2005. Solutions
Problem 1(40 p oints) Let A be the 3 5 matrix
2 1 1 1 3
A = 1 0 1 2 1 .
3 1 2 5 2
a)(20 p oints) Find all its right inverses, if
Row reduce the augmented matrix
2 1 1 1 3 | 1
1 0 1 2 1 | 0
3 1 2 5 2 | 0
they exist.
00
spancfw_w1 , . . . , wm , y = spancfw_w1 , . . . , wm , wm+1 .
R 1 2 4 2 1 F F + H O F K44 2 D H D : - O
.5r53GTG.cfw_G>53cfw_90Q52
w1 , w2 , . . . , wm , wm+1
POFHV- v - +H 1- +
QGQST0.cfw_#.,*
j =1
wm+1 = y
m
y, wj
wj .
wj , wj
- V 2 x P H V D R F
EXAM 1
Instructions: You will have approximately 50 minutes for this exam. The test is closed book,
closed notes and calculators are not allowed. The point value of each problem is written next to the
problem use your time wisely. Partial credit will be g
EXAM 2
Instructions: You will have approximately 50 minutes for this exam. The test is closed
book, closed notes and calculators are not allowed. The point value of each problem is written
next to the problem use your time wisely. Partial credit will be g
18.700 Problem Set 6 Solutions
1. (8 points) Suppose that we are given three polynomials
p2 (x) = ax2 + bx + c,
p1 (x) = dx + e,
p0 (x) = f
with real coecients. This problem is about the dierential operator
D = p2 (x)
d2
d
+ p1 (x)
+ p0 (x).
2
dx
dx
a) Ex
18.700 Problem Set 5 solutions
1. (6 points) Suppose we are given
three distinct elements x1 , x2 , and x3 in F ;
()
three arbitrary elements a, b, and c in F ;
()
and
The problem is to nd all polynomials
p(x) = u0 + u1 x + u2 x2 + u3 x3
( )
of degree les
18.700 Problem Set 3 Solutions
1. (3 points) Give an example of
whose reduced row-echelon form is
10
0 1
00
a 3 3 matrix A of real numbers
1/2
1/3
0
and such that every entry of A is a nonzero integer.
To get from A to the given row-echelon matrix, were
18.700 Problem Set 2 Solutions
1. (3 points) Let V be the vector space of polynomials of degree at most ve with
real coecients. Dene a linear map
T : V R3 ,
T (p) = (p(1), p(2), p(3).
That is, the coordinates of the vector T (p) are the values of p at 1,
18.700 Problem Set 4
For the rst problems, you may use the theorem I stated in class Tuesday October
8: there is a one-to-one correspondence
U Row(A)
between r -dimensional subspaces of F n and r n reduced row-echelon matrices
having exactly one pivot in
18.700 Problem Set 9 solutions
1. (5 points) Suppose V is a real or complex inner product space.
A linear map S L(V ) is called skew-adjoint if S = S . Suppose V is
complex and nite-dimensional, and S is skew-adjoint. Show that the
eigenvalues of S are al
18.700 Problem Set 7 Solutions
1. (6 points) Suppose that V is a complex inner product space with
orthogonal basis (f1 , . . . , fn ), and T L(V ).
a) Prove that any vector v V can be written
n
v=
i=1
v , fi
fi .
fi , fi
This essentially appears in the no
Proof of the spectral theorem
November 5, 2013
1
Spectral theorem
Here is the denition of selfadjoint, more or less exactly as in the text.
Denition 1.1. Suppose V is a (real or complex) inner product space. A
linear transformation S L(V ) is selfadjoint
18.700 Problem Set 8 solutions
1. (12 points) Suppose that V is a (real or complex) inner product
space, and that (t1 , . . . , tn ) is a basis of V .
a) Show that there is just one n n matrix U = (uij ) with the following
properties:
i) U is upper triang
Finite elds
I talked in class about the eld with two elements
F2 = cfw_0, 1
and weve used it in various examples and homework problems. In these notes I
will introduce more nite elds
Fp = cfw_0, 1, . . . , p 1
for every prime number p. Ill say a little ab
Orthogonal bases
The text emphasizes orthonormal lists, in keeping with tradition. Remember that a list (e1 , . . . , em ) is
called orthonormal if
ei , ej = 0,
(1 i = j m)
ON L
ei , ei = 1,
(1 i m).
These lists are indeed convenient, and lead to lots of