2. Matrices and Matrix Operations
2.1 Matrices, addition, subtraction and multiplication by a number
2.1.1 Matrices
A matrix is a rectangular array of numbers. It is what we frequently call a table of
numbers.
Example 2.1.1. (a continuation of Example 1.3
3.4 Electrical networks
One application of linear equations is to electrical networks containing voltage sources
and resistors. Here is a simple electrical network.
Example 1. This network consists of three lines that are connected at two nodes. The
posit
1.3 Linear Functions
A linear function is one where the variables are multiplied by numbers and then added up
or one that is equivalent to a function of this form. Any other function is called a nonlinear function. For example,
y = 7x
w = 2x 4y + 7z
y = 4
2.4 Geometric Transformations
In the previous section we discussed how to multiply matrices. One situation where this
is useful is with geometric transformations. Geometric transformations are rigid motions
of the plane. We shall consider some that leave
3 Linear Equations
This chapter is concerned with linear equations. We concentrate on two aspects to this.
One is transforming problems in the real world into linear equations. The other is on the
solution of linear equations.
3.1 Gaussian Elimination
Gau
1.4 Multiplication of vectors
1.4.1 Multiplying a row vector times a column vector
If a = (a1, a2, , an) is a row vector and x = is a column vector with the same number of
components then the product ax of a and x is simply the sum of the products of
corr
1.2 Vector Operations
1.2.1. Addition and subtraction
Numeric vectors. One adds and subtracts numeric vectors by adding and subtracting
corresponding components. In order to do this they have to have the same number of
components.
v=
w=
v+w=
v-w=.
For ex
1. Vectors and Vector Operations
1.1 Types of vectors
Three types of vectors that we will be concerned with are the following.
1.
2.
3.
numeric vectors lists of numbers
geometric / physical vectors things with magnitude & direction, e.g. directed
line seg
2.2
Multiplication of Vectors by Matrices Linear Functions
In section 1.4 we discussed how to multiply a row vector a times a column vector x to get
a number ax. We saw that by holding one of the two vectors a or x fixed, we obtained a
linear function of
2.3 Multiplication of matrices
In the previous section we discussed how to multiply a matrix by a vector. In this section
we consider multiplying two matrices A and B to form the matrix AB. There are three
equivalent ways that one may do this.
Method 1. O