Winter 2016 Syllabus
MATH201-011
Introduction to Statistical Methods I
(3 Credits)
Course Description:
MATH201 Basic topics in statistics will include exploratory data analysis, basic probability, discrete and continuous
distributions, sampling distributi
2016 Spring MATH 121 Final Exam Review
1. Perform the indicated operations.
Add, multiply and divide rational expressions.
Key: Use the following rules first:
(a)
(b)
(c)
A
D
A
B
A
B
B
A+B
D = D
C
AC
D
= BD
C
D
= AD
BC
+
then simply them by factoring the
College Algebra
Practice Section 2.7
1. Find an equation for the functions after following transformations.
(a) f (x) = x2 ; shift right 2 units, shift downward 3 units.
(b) f (x) = 3x 3; shift left 3 units, shift upward 1 unit.
(c) f (x) = x; shift right
College Algebra
Practice Section 3.2. and 3.3.
1. Determine if the function is a polynomial. If it is, state its degree and leading coefficient.
(a) f (x) = 2x3 x + 5
(b) 2x3
3
x
(c) 1 4x 51 x7
(d)
1
x2 +x+1
2. Divide the following
(a)
3x4 2x2 1
3x3
(b)
College Algebra
Practice Section 2.1.
1. Graph the following points in the xyplane.
P(-2,2),Q(4,0),R(5,-3),S(0,-3), and T(-2,-3).
2. Find the distance between points P(-2,5) and Q(-4,1). Find the midpoint of the line segment whose
end points are these poi
College Algebra
Practice Section 2.4.
1. Determine whether the following graphs represents a function or not.
2. Does the equation x4 + y 4 = 1 determine a function?
3. Let f (x) =
x + 1.
(a) Evaluate f (0), f (1), f (1), f (x 1),
(b) Find the domain of f
College Algebra
Practice Section 2.9
1. Determine whether the following functions are one-to-one:
(a) f (x) = (x 1)2
(b) g(x) = 3x + 1
2. Assume f is a one-to-one function.
(a) If f (3) = 12, find f 1 (12).
(b) If f 1 (4) = 9, find f (9).
3. The values of
College Algebra
Practice Section 2.2.
1. Determine whether (3, 4) is on the graph of the equation y = x 1.
2. Determine whether (3, 2) is on the graph of the equation
x + 1.
3. Graph each equation:
(a) y = x + 1
(b) y = x2
(c) |x + 1|
(d) x3 + 1
4. Find t
College Algebra
Practice Section 2.8
1. The values of f (x) and g(x) are given in the following tables. Evaluate the following based on the
tables.
x
0 1 2 3 4
f (x) 1 4 3 0 2
x
g(x)
0
0
1
4
2
2
3
2
4
1
(a) (f f )(1)
(b) (f g)(2)
(c) (g f )(4)
(d) (f g)(2
College Algebra
Practice Section 3.1
1. Find the standard form of the quadratic function f whose graph has vertex (1, 5) and passes through
the point (3, 7). Does f have a maximum or minimum value?
2. Graph the quadratic function f (x) = 2(x + 1)2 + 3. Fi
Chapter 6
Vector Spaces
6.1
Vector Spaces and Subspaces
x
. V is the set of all vectors in R2 whose first and second components are the same.
x
We verify all ten axioms of a vector space:
x+y
y
x
y
x
V since its first and second components are the
Chapter 4
Eigenvalues and Eigenvectors
4.1
Introduction to Eigenvalues and Eigenvectors
1. Following example 4.1, to see that v is an eigenvector of A, we show that Av is a multiple of v:
1
3
01+31
0 3 1
= 3v.
=3
=
=
Av =
1
3
31+01
3 0 1
Thus v is a
Chapter 3
Matrices
3.1
Matrix Operations
1. Since A and D have the same shape, this operation makes sense.
3 6
3+0 06
0 6
3 0
0 3
3 0
=
=
+
=
+2
A + 2D =
5
7
1 4 5 + 2
4
2
1 5
2
1
1 5
2. Since D and A have the same shape, this operation makes sens
3.5
Subspaces, Basis, Dimension, and Rank
Definition:
A subset W of a vector space V is called a subspace of V if W itself is a vector
space under the addition and scalar multiplication defined on V .
a) If u and v are vectors in W , then u + v is in W .
Test 2 Review MATH 349
This is a general list of topics that you should use while studying for Test 2. This test will
cover the sections below. You should use this list as a guide while you study the suggested HW,
notes, etc. Not everything on this list i
6.3
Change of Basis
Definition:
If we change the basis for a vector space V from an old basis B = cfw_u1 , u2 , . . . , un
to a new basis C = cfw_v1 , v2 , . . . , vn then for each v in V ,
(v)B = P (v)C
where the columns of P are the coordinate vectors
6.1
Vector Spaces and Subspaces
Definition: Vector Space Axioms
Let V be an arbitrary nonempty set of objects on which two operations are
defined: addition, and multiplication by scalars. By addition we mean a rule for
associating with each pair of object
5.2
Orthogonal Complements and Orthogonal Projections
Definition:
Let W be a subspace of Rn . We say that a vector V in Rn is Orthogonal to W
if v is orthogonal to every vector in W . The set of all vectors that are orthogonal
to W is called the Orthogona
3.3
The Inverse of a Matrix
Definition: Inverse
If A is a square matrix, and if a matrix B of the same size can be found such
that AB = BA = I, then A is said to be invertible (or nonsingular) and B is
called the inverse of A.
If such a matrix B cannot be
3.2
Matrix Operations
Properties of Matrix Arithmetic:
1. A + B = B + A
(commutative addition)
2. A + (B + C) = (A + B) + C
3. A(BC) = (AB)C
(associative addition)
(associative multiplication)
4. A(B C) = AB AC
(left distributive)
5. (B C)A = BA CA
(right
3.1
Matrix Operations
Definitions:
A Matrix is a rectangular array of numbers called Entries, or Elements, of
the matrix.
A general m n matrix has the form:
a11 a12
a
21 a22
.
.
.
.
am1 am2
a1n
a2n
. . . .
amn
This can be written more compactly
6.5
The Kernel and Range of a Linear Transformations
Definitions:
Let T : V W be a linear transformation.
The Kernel of T , denoted ker(T ), is the set of all vectors in V that are mapped
by T to 0 in W . That is,
ker(T ) = cfw_v V : T (v) = 0
The Range
6.4
Linear Transformations
Definition:
If T : V W is a function from a vector space V to a vector space W , then T
is called a linear transformation from V to W if the following hold for all u and v
in V and all scalars c.
1. T (u + v) = T (u) + T (v)
2.
3.4
The LU Factorization
Definition:
Let A be a square matrix. A factorization of A as A = LU , where L is unit
lower triangular and U is upper triangular is called an LU Factorization of A.
A Unit Lower Triangular matrix has 1s along the main diagonal an
5.3
The Gram-Schmidt Process and the QR Factorization
The Gram-Schmidt Process: Orthogonalization of a basis
To convert any basis cfw_x1 , x2 , . . . , xk into an orthogonal basis cfw_v1 , v2 , . . . , vk :
Step 1:
v1 = x1
Step 2:
v2 = x2
hx2 , v1 i
v1
Test 3 Review MATH 349
This is a general list of topics that you should use while studying. This test will cover the
sections below. You should use this list as a guide while you study the suggested HW, notes, etc.
Not everything on this list is guarantee
Test 4 Review MATH 349
This is a general list of topics that you should use while studying. This test will cover the
sections below. You should use this list as a guide while you study the suggested HW, notes, etc.
Not everything on this list is guarantee