Math 2B, Exam 1, Fall 2015
Name: _
Write all answers on scantron. You may write on this exam; write your name on it also, you well get it back!
(1) What is the row 2, column 1 entry in the transpose of the product
(A) 6a + 4b + 2c
(B) 6d + 4e + 2f
(C) 5a
1 [24 pts] TNE’orHE oJustification is required.
F; rN5; a [email protected] rix A has 6 di igenvalues, then the rank of A must be at least 5.
a L
[e f\
j ' ectors of A
Ff ' ' ' ere must be a baSIs, conSIstIng of eIgenv
b I: If matrIx A Is Mme, then th
2 x 2 matr
. .
MATH-23 Test 2 (Sec. 4.1-4.8) (100 points) Name: 0
1. [24 pts.]xi5rove or disprove (use theorems/results from text and lecture) '
a) The pelynomiais x~1, ( 1I)2_, and 7
r' I ,
7"} -"' xi ' . ' i,
4 I F - _. \ j. f: .
Ifin vector space t e set of v
MATH-2B
Project (part 1) (25 points)
Name:_
Directions: Read the assigned material and provide the complete solutions of the following two
problems on separate sheets of paper.
Due date: Monday, July 27, at the beginning of the class.
Problem 1 (15 pts.)
MATH-2B
Sec. 6.3
Gram-Schmidt Process; QR-Decomposition
Def 1 A set of two or more vectors in a real inner product space is said to be orthogonal if all pairs of
distinct vectors in the set are orthogonal. An orthogonal set in which each vector has norm 1
MATH-2B
Sec. 6.1
Inner Products
Def 1 An inner product on a real vector space
u, v
with each pairs of vectors
for all vectors
1.
,
u v
, and
w
in
u
V
and
v , v 0
and
in such a way that the following axioms are satisfied
k
.
[Additivity axiom]
[Homogeneity
MATH-2B
Sec. 5.3
DEFINITION 1 If
numbers
Complex Vector Spaces
n is appositive integer, then a complex n -tuples is a sequence of n complex
z1 , z 2 ,., z n . The set of all complex n -tuples is called complex n-space and is denoted by C n .
Scalars are
MATH-2B
Sec. 5.2
Diagonalization
The Matrix Diagonalization Problem
Our first objective in this section to show that the following two seemingly different problems are
equivalent.
Problem 1 Given an
diagonal?
n n matrix A , does there exist an invertible
MATH-2B
Sec. 5.1
Eigenvalues and Eigenvectors
DEFINITION 1 If A is an n n matrix, then a nonzero vector x in R n is called an eigenvector of A (or of
the matrix operator T A ) if Ax is a scalar multiple of x , that is
Ax x
for some scalar . The scalar is
Math-2B
Sec. 7.1
Def 1 A square matrix
Orthogonal Matrices
A
is said to be orthogonal if its transpose is the same is the same as its
inverse, that is,
(1)
A 1 AT
or, equivalently, if
AAT AT A I
(2)
Ex 1 A 3x3 Orthogonal Matrix
(a)
1
3
2
A
3
2
3
2
3
2
MATH-2B
Quiz 5 (Sec 4.7)
Directions: Show all your work.
1. [4 pts.]
Name:_
Total: 16 pts.
Find a null space for
A
.
2 0 1
A 4 0 2
0 0 0
2. [8 pts.] Use the methods of Ex 6 and Ex 7 in Sec. 4.7 to find a basis for the raw space and the
column space of th
MATH-2B
Sec. 6.2
Angle and Orthogonality in Inner Product Spaces
Th 6.2.1 Cauchy-Schwartz Inequality
If
u
and
v
are vectors in a real inner product space
V
, then
u, v u v
The alternative forms of Cauchy-Schwartz inequality:
u, v
u, v
2
2
u , u v, v
u
We
MATH-2B
Quiz 6 (take home) Ch 5
Directions: Completely solve the following problems on separate sheets of paper.
Due date: Tuesday, July 28, at the beginning of the class.
Sec. 5.1 #3(b), 4(b),5(b); 6(d), 7(d), 8(d); 12(b), 14, 18, 26, 28, T-F(b,c,f).
Sec
1)
QUIZ #2 - SOLUTIONS
T
T
Is w = ( 7,8,9 ) a linear combination of u = (1,2, 3) and
T
v = ( 4,5,6 ) ? (Show work).
2)
1 4 7 rref 1 0 1
2 5 8 0 1 2
3 6 9 0 0 0
Yes. w = u + 2 v
L
1)
QUIZ #3 - SOLUTIONS
Use Gram-Schmidt to transform the basis vectors
1 3
,
into an orthonormal basis using the standard
0 5
Euclidean inner product.
1
u1 =
0
2)
u2 =
3
5
3
5
MATH-2B
Sec. 1.5 and Sec. 1.6
Sec. 1.5 Elementary Matrices and a Method for Finding A 1 .
Definition 1 Matrices A and B are said to be row equivalent if either (hence each) can be
obtained from the other by a sequence of elementary row operations.
Our nex
MATH-2B
Sec. 7.3
Quadratic Forms
Definition of a Quadratic Form
Expression of the form
an
a1 x1 a 2 x 2 a n x n
in our study of linear equations. If
,
a1 a 2
,.,
are treated as fixed constants, then this expression is called a linear form and it a real-va
MATH-2B
Review for Test 3
1. True-False Questions on eigenvalues, eigenvectors, the characteristic equation (or the
characteristic polynomial), the characteristic roots, an orthogonal complement, recognizing of
linear operators from a formula for transfor
MATH-2B
Review for Final Exam
1. Main facts about system of linear equations, including rewriting the system as a matrix equation;
basis of the column space, the row space, and the null space of a matrix.
2. Construction of the basis of the orthogonal com
MATH-2B
Def 1 If
Sec. 7.2
A
and
B
Orthogonal Diagonalization
are the square matrices, then we say that
If there is an orthogonal matrix
If
A
P
such that
P T AP B
and
B
are orthogonally similar
.
is orthogonally similar to some diagonal matrix, say,
(1)
T
MATH-2B
Sec 6.4 Best Approximations; Least Squares
Least Square Problem
Given a linear system
Ax b
of
m
equations in
with respect to the Euclidean inner product on
system, we call
b Ax
n
R
unknowns, find a vector
m
. We call such an
the least square error
MATH-2B
Sec. 4.8 Rank, Nullity, and the Fundamental Matrix Spaces
Let consider row echelon matrix
1
0
R
0
0
2
0
0
0
3 4
0 1
0 0
0 0
0
2
0
0
2
7
1
0
Since the rows containing leading 1s (nonzero rows) form a basis for the row space and the columns
contain
MATH-2B
Sec. 4.6 Change of Basis
Coordinate Maps
If S v1 , v 2 ,., v n is a basis for a finite-dimensional vector space V , and if
v S c1 , c 2 ,., c n
Is the coordinate vector of
v relative to
(1)
S
creates a one-to-one correspondence between vectors i
MATH-2B
Sec. 4.2
Subspaces
It is possible for one vector space to be contained within another. We will explore this idea in this
section, we will discuss how to recognize such vector spaces, and we will give a variety of examples that
will be used on our
MATH-2B
Quiz 6 (take home) Ch 5
Directions: Completely solve the following problems on separate sheets of paper.
Due date: Friday, June 3, at the beginning of the class.
Sec. 5.1 #3(b), 4(b),5(b); 6(d), 7(d), 8(d); 12(b), 14, 18, 26, 28, T-F(b,c,f).
Sec.
MATH-2B
Homework 3
Due date: to be announced.
Sec. 2.1: #10, #28
Sec. 2.2: #12, #28
Sec. 2.3: #12, #26, #30
Sec. 3.1: #20, #22a, #28
Sec. 3.2: Prove Theorems 3.2.5, 3.2.6, and 3.2.7; #20a, #22, #26a, T-F (j)
Sec. 3.3: #5, #10, #14, #20, #24, #34, #38
Sec.
MATH-2B
Homework 6
Directions: Show complete solutions for the assignment below on separate sheets of paper.
Due date: Thursday, June 9, at the beginning of the class.
Assignment:
Sec. 6.1: #4(a,b), 8(a)
Sec. 6.2: #1(e), 5, 6, 8, 14
Sec. 6.3: 5(a), 14(b),
MATH-2B
Project (part 2)
(25 points)
Name:_
Directions: Please, read sec. 9.3 (Internet Search Engines) carefully and write a complete solution of
exercise #6 from Exercise Set 9.3 (page 501) using the appropriate concepts and terminology.
Due date: Frida