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Lecture 5: Bases, Generating and linear independent systems We are going to use the vector spaces R2
 

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}

  

Source: Abbas, Casim - Department of Mathematics, Michigan State University

 

Collections: Mathematics