Intro
Translations in the plane
Examples
HowTo: Gaussian Reduction2.1. Linear (in)dependence.
We start with a definition.
Definition 2.1. A set 
of vectors in 
is linearly dependent
if there exist scalars 
not all zero such that
is said to
be linearly independent. Equivalently, 
is
linearly independent if and only if
Problem 2.2. Let 
be a linearly independent set. Prove that if
then 
for all 
.
Example 2.3. In the plane, any three or more vectors form
a linearly dependent set, whereas any set consisting of one nonzero vector
or any set consisting of two non-collinear vectors is linearly independent.
The same holds in 
. In 
four or more
vectors are linearly dependent, whereas any two non-collinear vectors or any
three non-coplanar vectors are linearly independent.
Problem 2.4. Any set containing the zero vector 
is linearly dependent. Any set containing two vectors, one of which is a multiple
of the other, is linearly dependent.
Problem 2.5. Show that a set consisting of two vectors
is linearly dependent if and only if one vector is a scalar multiple of the
other. Is the subset of 
given by
linearly dependent?
Example 2.6. In 
,
the set 
with 
is linearly
independent, where the 
are elementary vectors.
