6-22.md

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title: 6/20

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Plan

• 1.5

• 1.6

Remarks from 1.4

• 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 $u_1,u_2,\ldots,u_n$ in $S$ and scalars $a_1,\ldots,a_n$, not all zero, such that $a_1u_1+\cdots+a_1u_n=0$.

• 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 $u\in S$ such that $u\in span(S\setminus \{u\})$.

• (Spans don't grow if you add linearly dependent things.) Theorem: Let $S$ be a subset of a vector space $V$ and $A\subseteq span(S)$. Then $span(S)=span(S\cup A)$.

• (Linearly dependent means you have redundant elements) Corollary.

• Example: Give polynomial example

1.6 Basis

• Definition: Basis....

• Examples