De Casteljau and Bezier Curves
Definition: An indexed set of vectors cfw_v1, , vp in
is said to be linearly independent if the vector
x v + x v + . + x v =
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
A matrix with only one column is called a column
vector, or simply a vector.
An example of a vector with two entries is
w = ,
where w1 and w2 are any real numbers.
The set of all vectors with 2 entries is denoted b
A rectangular matrix is in echelon form (or row
echelon form) if it has the following three
1. All nonzero rows are above any rows of all
2. Each leading entry of a row is in a column to
the right of the leading entry of th
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
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.
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
Such a system Ax = 0 always has at least one
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
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
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
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
0 0 1 3 2
0 0 0 1 4
Continue the appropriate row operations and describe the solution set of
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
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
Applications Project I
1. Find the interpolating polynomial of degree 3 which passes through (1,29), (-1,-35), (2,31) and
p (t) = a0 + a1 * t + a2 * t 2 + a3 * t 3 1, 29 , - 1, - 35 , 2, 31 , - 3, - 19