This chapter introduces the notions of linear independence and span. Now instead of worrying about single vectors and how they interact or worrying about small lists of vectors and vector spaces, we start worryihng about properties of lists of vectors.
These ideas are closely related to vector spaces, sums of vector spaces, and subspaces.
Important Definitions A linear combination of a list \({latex.inline[v_{1}, ..., v_{m}](v_{1}, ..., v_{m})} of vectors in V iss a vector of the form \){latex.inlinea{1}v{1} + ... + a{m}v{m}} where ${latex.inlinea{1}, ..., a{m} \in F}. Note that this is a bit similar to the proces we use to take the sum of vector spaces(albeit with one vector from each space).
The set of all linear combinations of a list of vectors \({latex.inline[v_{1}, ..., v_{m} \in V](v_{1}, ..., v_{m} \in V)} is called the Span of \){latex.inlinev{1}, ..., v{m}}.
A vector space is called finite dimensional if it is spanned by a list of vectors that is of finite length. A vector space is called infinite dimensional if it is not finite dimensional.
A list of vectors is called linearly independent if the only choice of \({latex.inline[a_{1}, ..., a_{m} \in F](a_{1}, ..., a_{m} \in F)} that makes \){latex.inlinea{1}v{1} + ... + a{m}v{m} = 0} is ${latex.inlinea{1} = ... = a{m} = 0}.
A vector that is not linearly independent is called linearly dependent.
Relevant Theorems * 1753232225 - Axler 2.6 Span is the smallest containing subspace * 1753232251 - Axler 2.19 Linear dependence lemma * 1753232272 - Axler 2.22 Length of linearly independent list <= length of spanning list * 1753232272 - Axler 2.25 Finite dimensional subspaces