Table of Contents
#source Axler2.A
1 #definition span
The set of all linear combinations of a list of vectors \(v_1, ..., v_m\) in \(V\) is called the span of \(v_1, ..., v_m\), denoted \(\text{span}(v_1,...,v_m)\): \[\text{span}(v_1,...,v_m) = {a_1v_1 + ... + a_mv_m | a_1, ..., a_m \in F}\] And the span of an empty list \(()\) is \({0}\) - This is just to make Axler2.C work out nicely (KBeRefLinAlgDimension)
2 Properties
- The span is the smallest containing subspace
The span of a list of vectors in \(V\) is the smallest subspace of \(V\) containing all the vectors in the list.
2.1 #definition spans
If \(\text{span}(v_1,...,v_m) = V\), then \(v_1, ..., v_m\) spans \(V\)
3 Examples
3.1 Axler 2.9
Suppose \(n\) is a positive integer. Show that \((1, 0, ..., 0), (0, 1, 0, ..., 0), ..., (0, ..., 0, 1)\) spans \(F^n\). - Basically, if a list of vectors spans a vector space then linear combinations of those vectors (almost like colloquial polynomials of those vectors) can form each vector in the space. - In this case, the vector space \(F^n\) is a list of vectors in \(F\), and having the \(1\) in each slot is enough to, when scalar multiplied with \(a \in F\), get all possibilities of \(F^n\). - I need to wrap my head around this some more.