1.3 The Algebra of Vector Operations
In the previous two sections we looked at row and column vectors, matrices, functions
and directed line segments all of which are called vectors. For each of these types of
vectors we saw how addition, negation, subtra
1.6 Multiplication of matrices
In the previous section we discussed the product of a matrix and a vector and we saw that
this product allowed us to specify all linear functions that map column vectors to column
vectors. In this section we consider multipl
1.4 Multiplication of Vectors and Linear Functions
The previous three sections were concerned with vectors and vector operations. The next
three sections look at multiplication of vectors and matrices and linear functions which
are another important ingre
4.4 Applications of Determinants
In this section we look at applications of determinants to solving equations and to
volumes.
Solving Equations with Determinants (Cramer's Rule). Consider a system of n
equations with n unknowns.
a11x1 + a12x2 + + a1nxn =
2.2 Gaussian Elimination with Row Exchanges.
In the previous section we discussed Gaussian elimination. In that discussion we used equation 1 to
eliminate x1 from equations 2 through n. Then we used equation 2 to eliminate x2 from equations 2 through
n an
1.5 Multiplication of Vectors by Matrices
In the previous section we saw that a real valued linear function z = f(x) of a column
vector x = had the form f(x) = ax where a = (a1, a2, , an) is a row vector. In this section
we extend multiplication to a matr
1. Vectors and Linear Functions
We begin with vectors and linear functions which provide the framework for the problem
solving methods of linear algebra.
1.1 Vectors
There are various types of objects to which the term vector is applied. These include row
1.2 Vector Operations
Row and column vectors, matrices, functions and directed line segments are all regarded
as vectors because they can be added, negated and multiplied by numbers. Subtraction of
vectors is sometimes regarded as a secondary operation be
2
Linear Equations
In this chapter we look at solving linear equations and inverses of matrices and linear functions. In the case
where the unknown is a row or column vector, Gaussian elimination is a efficient procedure to solve linear
equations. As we s
Exam 2 Math 513 Winter 1995
Name: This is a closed book exam. You my E
use a calculator. Show yetJr work and explain any reasoning which is not
clear frcm the ccnputations. This is particularly iuportant for me to be able
to give you part credit if you