Plan

• 1.5
• 1.6

Remarks from 1.4

• In this class, we allow spans of infinite sets.
• Example, what is the span of xⁿ 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 u₁, u₂, …, u_n in S and scalars a₁, …, a_n, not all zero, such that a₁u₁ + ⋯ + a₁u_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 ∈ span(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 ⊆ span(S). Then span(S) = span(S ∪ A).
• (Linearly dependent means you have redundant elements) Corollary.
• Example: Give polynomial example

1.6 Basis

• Definition: Basis….
• Examples