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} Collections: Mathematics