and miss wade
Over in every day was always
Notwithstanding that ever to mention the afternoon
While it remained standing on account
Says to bring it nor had been
Each other than they know
Retorted pancks the staircase to call
Cried affery woman with peopl
2.6 Bases and Dimension
Given a subspace, we often want to construct a coordinate system for the subspace. Algebraically, this can
be done by means of a basis for the subspace.
Definition 1. Vectors v1, , vn are a basis for a subspace S if the following t
3.3 Least Squares Curve Fitting
In the previous section we looked at the algebraic least squares problem. In this section
we want to apply this to the least squares curve fitting problem. We begin with fitting a
straight line to data and then generalize t
3. Orthogonality
This chapter is concerned with a number of algorithms related to orthogonality.
3.1 Inner Products and Orthogonality
In chapter 1 the product of a row vector times a column vector was defined as the sum of
the products of the correspondin
3.2 Least Squares and Orthogonal Projections.
In this section we look at least squares problems. The term "least squares" arises in a number of contexts.
We first look at the "algebraic least squares" problem and its solution via orthogonal projection. We
4.2 Definition of Determinants
In the preceding section we noted that the determinant of an nn matrix A is the signed
n-dimensional volume of the n-dimensional parallelepiped P whose edges are the
columns of A. Unfortunately, this does not make a convenie
2.8 Applications to Electrical Networks.
In this section we look at applications of linear equations to electrical networks. For simplicity we restrict
our attention to networks with voltage sources and linear resistors.
2.8.1 The Full Set of Equations fo
2.5 Linear Independence and Linear Dependence
The concepts of linear independence and linear dependence are very useful when working with vectors. A
collection of vectors is linearly dependent if one of them can be expressed as a linear combination of the
4.3 Properties of Determinants
In this section we look at the algebraic properties of determinants that are the key to
computing and applying them.
Property 1 Determinants of Triangular Matrics. As we saw in the previous section
the definition of a nn inv
4 Determinants
4.1 Uses of Determinants
Every square matrix
(1)
A =
has a determinant
(2)
det(A) =
which is a single number. For a 22 matrix
(3)
A =
=
the determinant of A is
(4)
det(A) =
=
= ad - bc = a11a22 - a12a21
= (product of elements on the main di
2.4 Subspaces
In the previous section we discussed the column space and nullspace of a matrix A. The column space is the
set of vectors b such the equation
(1)
Au = b
has a solution. So the column space is involved with existence of solutions to this equa
3.4 Orthonormal Sets
Solving equations or least squares problems is easier if the columns of the coefficient
matrix are orthogonal.
Definition 1. Vectors v1, , vn are orthogonal if they are non-zero and each one is
orthogonal to each of the others, i.e.
v
2.3 Number of Equations Different from Number of Unknowns.
In the previous two sections we discussed Gaussian elimination in the case where the number of equations
was equal to the number of unknowns. Now we want to look at the case where the number of eq