Math 120 Final Exam Review
Determine whether the relation is a function.
1) cfw_(-7, -4), (-5, -7), (-2, -5), (2, -6)
A) Not a function
B) Function
Determine whether the equation defines y as a function of x.
2) y2 = 4x
A) y is a function of x
B) y is not
Warm-Up Exercises:
De Casteljau and Bezier Curves
Quadratic Case:
Cubic Case:
Section 1.7
LINEAR INDEPENDENCE
Definition: An indexed set of vectors cfw_v1, , vp in
n
is said to be linearly independent if the vector
equation
x v + x v + . + x v =
0
1
1
2
Linear Algebra
Vector Spaces (finite or infinite dimensional)
Linear mappings between vector spaces
Study of which is motivated by system of linear equations
Such equation are naturally represented using matrices and vectors.
Linear Algebra is central to
VECTOR EQUATIONS
Vectors in
A matrix with only one column is called a column
vector, or simply a vector.
An example of a vector with two entries is
w1
w = ,
w2
where w1 and w2 are any real numbers.
The set of all vectors with 2 entries is denoted b
ECHELON FORM
A rectangular matrix is in echelon form (or row
echelon form) if it has the following three
properties:
1. All nonzero rows are above any rows of all
zeros.
2. Each leading entry of a row is in a column to
the right of the leading entry of th
Warm-Up Exercises:
MATRIX OPERATIONS
If A is an m n matrixthat is, a matrix with m rows
and n columnsthen the scalar entry in the ith row
and jth column of A is denoted by aij and is called the
(i, j)-entry of A. See the figure below.
Each column of A i
DEFINITION
DEFINITION
THEOREM 1
DEFINITION
THEOREM 2
DEFINITION
THEOREM 3 The column space of an m x 11 matrix A is all of IR if and only if the equation
Ax = b has a solution for each h in R.
DEFINITION
THEOREM 4
DEFINITION
THEOREM 5
THEOREM 6
THEO
Warm-Up Exercises:
HOMOGENEOUS LINEAR SYSTEMS
A system of linear equations is said to be
homogeneous if it can be written in the form Ax = 0 ,
where A is an m n matrix and 0 is the zero vector in
m.
Such a system Ax = 0 always has at least one
solution
LINEAR COMBINATIONS
Given vectors v1, v2, ., vp in n and given scalars c1,
c2, ., cp, the vector y defined by
y= c1v1 + . + c p v p
is called a linear combination of v1, , vp with
weights c1, , cp.
The weights in a linear combination can be any real
numb
THE INVERTIBLE MATRIX THEOREM
Theorem 8: Let A be a square n n matrix. Then
the following statements are equivalent. That is, for
a given A, the statements are either all true or all
false.
a. A is an invertible matrix.
b. A is row equivalent to the n n
NULL SPACE OF A MATRIX
Definition: The null space of an m n matrix A,
written as Nul A, is the set of all solutions of the
homogeneous equation Ax = 0. In set notation,
Nul A cfw_x
=
: x is in n and Ax 0.
Theorem 2: The null space of an m n matrix A is
a
Integration by Parts
The purpose of this set of exercises is to show how the matrix of a linear transformation relative
to a basis B may be used to find antiderivatives usually found using integration by parts.
To find t 2et dt , the normal approach would
MATH 260 Final Exam Review Problems
1. The augmented matrix of a linear system has been reduced by row operations to the form
1 1 0 0 5
0 1 2 0 7
shown here:
0 0 1 3 2
0 0 0 1 4
Continue the appropriate row operations and describe the solution set of
MATRIX OPERATIONS
An n n matrix A is said to be invertible if there is
an n n matrix C such that
CA = I and AC = I
where I = I n , the n n identity matrix.
In this case, C is an inverse of A.
In fact, C is uniquely determined by A, because if B
were an
Part1: Interpolating Polynomails
The purpose of this set of exercises is to show how to use a system of linear
equations to fit a polynomail throught a set of points. This execise set expands ideas
began in problems 33 and 34 of section 1.2, and applies t
MATH 260
Applications Project I
Nadaly Marchi
Seth Killian
PART I
1. Find the interpolating polynomial of degree 3 which passes through (1,29), (-1,-35), (2,31) and
(-3,-19).
p (t) = a0 + a1 * t + a2 * t 2 + a3 * t 3 1, 29 , - 1, - 35 , 2, 31 , - 3, - 19
THE UNIQUE REPRESENTATION THEOREM
Theorem 7: Let = cfw_b1 ,., b n be a basis for
vector space V. Then for each x in V, there exists a
unique set of scalars c1, , cn such that
x= c1b1 + . + cn b n
-(1)
Proof: Since spans V, there exist scalars such that
(
ORTHOGONAL SETS
Thus S is linearly independent.
n
Definition: An orthogonal basis for a subspace W of
is a basis for W that is also an orthogonal set.
Theorem 5: Let cfw_u1,up be an orthogonal basis for a
n
subspace W of . For each y in W, the weights
Condition Numbers
The purpose of this set of exercises is to show how a condition number of a matrix A
may be defined, and how its value affects the accuracy of solutions to systems of
equations Ax = b.
Consider the following equation Ax = b :
1
2
3+
1 1
Equilibrium Temperature Distributions
The purpose of this set of exercises is to discuss a physical situation in which solving a
system of linear equations becomes necessary: that of determining the equilibrium
temperature of a thin plate. Methods for sol