 
Summary: Lecture 5: Bases, Generating and linear independent systems
We are going to use the vector spaces R2
and R3
to explain the concept of a
base and the concepts of a generating system and a linear independent system. We
identify R2
with arrows in the plane and R3
with arrows in three dimensional space
with the 'familiar' vector operations (meaning familiar from high school mathemat
ics, see also my notes for lecture #3). Given a bunch of vectors v1, . . . , vn in a
vector space V , we need to 'see' what the set of all linear combinations of the set
{v1, . . . , vn} looks like. We will use the notation span{v1, . . . , vn} for this set, i.e.
span{v1, . . . , vn} =
n
k=1
kvk V 1, . . . , n R
Let us start with a single nonzero vector v in the plane. A linear combination of v is
any vector of the form v where R is a real number. If we let run through all
the real numbers, we just change the length of v or its direction, hence span{v} is
a line through the origin. Let us now take two vectors: What about span{2v, v}
