Vectors & Scalars
Vectors
R2 is the set of all pairs of real numbers. In the context of drawing graphs,
the objects in R2 are called points, and pairs are written left-to-right, so that
(3, 2) is the point in R2 whose x-coordinate equals 3 and whose y-coo
Determinants & Inverse Matrices
The determinant of the 2 2 matrix
a b
c d
is the number ad cb.
The above sentence is abbreviated as
a b
det
= ad cb
c d
Example.
4 2
det
1 3
= 4(3) 1(2) = 12 + 2 = 10
The determinant of a 3 3 matrix can be found using the f
Rows & Columns
In this chapter well learn how to multiply a row of n numbers and a column
of n numbers to obtain a single number. Actually, in this class, n will always
be either 2 or 3, though it could be any natural number.
Rows
A row of 2 numbers is ju
Substitution
In this chapter, well examine systems of two linear equations in two variables that have unique solutions. If a system has a unique solution, we can
use a method called substitution to nd the unique solution.
How to nd the solution
Suppose yo
Matrix Equations
This chapter consists of 3 example problems of how to use a matrix equation to solve a system of three linear equations in three variables.
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Problem 1. Use that
1
1 2 1
1 1 0
2 2 1 = 1 2 1
3 1 1
4 5 2
to nd x, y,
2 2 Matrices
A 22 matrix (pronounced 2-by-2 matrix) is a square block of 4 numbers.
For example,
2 1
1 1
is a 2 2 matrix. Its called a 2 2 matrix because it has 2 rows and 2
columns. The four numbers in a 2 2 matrix are called the entries of the
matrix.
T
Linear Equations in Three Variables
Linear Equations Three Variables
Variables
Linear Equations in Three Variables
Linear Equations inin ThreeVariables
2
R is the space of 2 dimensions. There is an x-coordinate that can be any
dimensions. There is x-coord
Linear Equations in Two Variables
In this chapter, well use the geometry of lines to help us solve equations.
Linear equations in two variables.
If a, b, and r are real numbers (and if a and b are not both equal to 0) then
ax + by = r is called a linear e
Sets & Numbers
Sets
A set is a collection of objects. For example, the set of days of the week is
a set that contains 7 objects: Mon., Tue., Wed., Thur., Fri., Sat., and Sun.
Set notation. Writing cfw_2, 3, 5 is a shorthand for the set that contains the
n
3 3 Matrices
Much of this chapter is similar, to the chapter on 2 2 matrices. The most
substantial dierence between 2 2 matrices and 3 3 matrices is that its
harder to write a 3 3 matrix than it is to write a 2 2 matrix.
3 3 matrices have 3 rows and 3 col