## Plan

- In this class, we allow spans of infinite sets.
- Example, what is the span of
*x*^{n} inside the space of all functions?
- Vector spaces is a natural setting for linear transformations.
- Spanning and generating.

## 1.5 Linear independence

- Definition: A subset
*S* of a vector space *V* is called linearly dependent if there exists a finite number of distinct vectors *u*_{1}, *u*_{2}, …, *u*_{n} in *S* and scalars *a*_{1}, …, *a*_{n}, not all zero, such that *a*_{1}*u*_{1} + ⋯ + *a*_{1}*u*_{n} = 0.
- A subset
*S* is linearly independent if it is not linearly dependent.
- Theorem: A subset
*S* of a vector space *V* is linearly dependent if and only if there exists an element *u* ∈ *S* such that *u* ∈ *s**p**a**n*(*S* \ {*u*}).
- (Spans don’t grow if you add linearly dependent things.) Theorem: Let
*S* be a subset of a vector space *V* and *A* ⊆ *s**p**a**n*(*S*). Then *s**p**a**n*(*S*) = *s**p**a**n*(*S* ∪ *A*).
- (Linearly dependent means you have redundant elements) Corollary.
- Example: Give polynomial example

## 1.6 Basis

- Definition: Basis….
- Examples