Lecture Notes in Differential Equations - Bruce E. Shapiro

Lesson 14
Review of Linear Algebra
In this section we will recall some concepts from linear algebra class
Definition 14.1. A Euclidean 3-vector v is object with a magnitude
and direction which we will denote by the ordered triple
v = (x, y, z) (14.1)
The magnitude or absolute value or length of the v is denoted by the
postitive square root
v = |v| = √ x 2 + y 2 + z 2 (14.2)
This definition is motivated by the fact that v is the length of the line
segment from the origin to the point P = (x, y, z) in Euclidean 3-space.
A vector is sometimes represented geometrically by an arrow from the origin
to the point P = (x, y, z), and we will sometimes use the notation (x, y, z)
to refer either to the point P or the vector v from the origin to the point
P . Usually it will be clear from the context which we mean. This works
because of the following theorem.
Definition 14.2. The set of all Euclidean 3-vectors is isomorphic to the
Euclidean 3-space (which we typically refer to as R 3 ).
If you are unfamiliar with the term isomorphic, don’t worry about it; just
take it to mean “in one-to-one correspondence with,” and that will be
sufficient for our purposes.
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