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
. .
MATH23 Test 2 (Sec. 4.14.8) (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
MATH2B
Sec 3.4 The Geometry of Linear Systems
Lines in Space
Parametric Equations of a Line in Space
A line L parallel to the vector
v a, b, c
and passing through the point P x1 , y1 , z1 is represented by the
parametric equations
x x1 at
y y1 bt
z z1 ct
MATH2B
Sec. 4.5
Dimension
Number of Vectors in a Basis
THEOREM 4.5.1
All bases for a finitedimensional vector space have the same number of vectors.
This theorem follows immediately from the following
THEOREM 4.5.2 Let V be a finitedimensional vector s
MATH2B
Sec. 4.4
Coordinate and Basis
Coordinate Systems in Linear Algebra
Y
b
 P(a,b)
Coordinates of P in a rectangular coordinates system

in 2space
a
x
bv2
P(a,b)
v2
Coordinates of P in nonrectangular coordinate system
in 2space
O V1
av1
DEFINITION
MATH2B
Sec. 4.3
Linear Independence
We are looking for a minimal spanning set S of vectors in a vector space V . In order to be minimal,
any vector in S cannot be a linear combination of the others vectors in S . Indeed, if vector v in S is a
such linear
MATH2B
Test 1 (takehome portion)
Name:_
Directions: Show all work on separate sheets of paper.
Due date: Tuesday, July 14, at the beginning of the class.
1.
[15 pts.] Let consider a linear system.
x1 3 x 2 4 x3 6
3 x1 10 x 2 10 x3 3
2 x1 4 x 2 11x3 9
MATH2B
Sec. 4.1
Real Vector Space
In this section we will extend the concept of a vector by using the basic properties of vectors in R n as
axioms, which if satisfied by a set of objects, guarantee that those objects behave like familiar vectors.
Vector
MATH2B
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
MATH2B
Sec. 4.6 Change of Basis
Coordinate Maps
If S v1 , v 2 ,., v n is a basis for a finitedimensional vector space V , and if
v S c1 , c 2 ,., c n
Is the coordinate vector of
v relative to
(1)
S
creates a onetoone correspondence between vectors i
MATH2B
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
MATH2B
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 nspace and is denoted by C n .
Scalars are
MATH2B
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
MATH2B
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
Math2B
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
MATH2B
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
MATH2B
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
MATH2B
Sec. 9.1 LUDecomposition
Solving Linear Systems by Factoring
Our first goal in this section is to show how to solve a linear system
unknowns by factoring the coefficient matrix
A
L
is lower triangular and
U
into product
(1)
A LU
where
of equation
MATH—ZB Quiz 3 (Ch 2 and Ch 3) 16 pts.
4 5 0
1. For the matrix A = 1 1 1 2
1 5 2
a. Write down the minor M32 and evaluate it by hand.
M32: [Lil S:@)(2)~CH)(0329
b. Write down the cofactor C32 and evaluate it by hand.
Caaréllgul‘lseaeilWake1W): «.9
c. W
Math 2B, Exam 3, Fall 2015
Name: _
Open book, open notes, calculator allowed. No communication capable devices (laptops, cell
phones, tablets, Google glasses, etc.) allowed. Place all answers on scantron, and turn in
scantron
Math 2B, Fall 2015, Exam 2
Name: _
Open book, open notes, calculator allowed. No communication capable devices (laptops, cell phones, tablets,
etc.) allowed. Place all answers on scantron, and turn in scantron I
MATH2B
Sec. 1.3 Operations with Matrices
Representations of Matrices
1. A matrix can be denoted by an uppercase letters such as A , B , or C .
2. A matrix can be denoted by a representative element enclosed in brackets, such as a ij , bij or
c .
ij
3.
MATH2B
Sec. 1.1 Introduction to Systems of Linear Equations
Sec. 1.2 Gaussian Elimination
Matrices
In this section, we will study a streamlined technique for solving systems of linear equations. This
technique involves the use of a rectangular array of r
MATH2B
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
MATH2B
1.
Sec. 1.8 and 1.9
Applications
Finding a Parabolic Path through Three Points
A convenient way to draw curves of desired shapes is to select some points on the curve and find a
polynomial whose graph goes through these points. Two points, for exa
MATH2B
Sec. 1.7
Diagonal, Triangular and Symmetric Matrices
Diagonal Matrices
A square matrix in which all the entries off the main diagonal are zeros is called a diagonal matrix.
A diagonal matrix is invertible if and only if all of its diagonal entries
MATH2B
1.
Sec. 1.8 and 1.9
Applications
Finding a Parabolic Path through Three Points
A convenient way to draw curves of desired shapes is to select some points on the curve and find a
polynomial whose graph goes through these points. Two points, for exa