# 12-240/Classnotes for Tuesday October 09

In this lecture, the professor concentrated on bases and related theorems.

## Definition of basis

β ${\displaystyle \subset \!\,}$ V is a basis if

1/ It generates (span) V, span β = V

2/ It is linearly independent

## Theorems

1/ β is a basis of V iff every element of V can be written as a linear combination of elements of β in a unique way.

proof: ( in the case β is finite)

β = {u1, u2, ..., un}

(<=) need to show that β = span(V) and β is linearly independent.

The fact that β span is the fact that every element of V can be written as a linear combination of elements of β, which is given

Assume ${\displaystyle \sum \!\,}$ ai∙ui = 0 ai ${\displaystyle \in \!\,}$ F, ui ${\displaystyle \in \!\,}$ β

${\displaystyle \sum \!\,}$ ai∙ui = 0 = ${\displaystyle \sum \!\,}$ 0∙ui

since 0 can be written as a linear combination of elements of β in a unique way, ai=0 ${\displaystyle \forall \!\,}$ i

Hence β is linearly independent

(=>) every element of V can be written as a linear combination of elements of β in a unique way.

So, suppose ${\displaystyle \sum \!\,}$ ai∙ui = v = ${\displaystyle \sum \!\,}$ bi∙ui

Thus ${\displaystyle \sum \!\,}$ ai∙ui - ${\displaystyle \sum \!\,}$ bi∙ui = 0

${\displaystyle \sum \!\,}$ (ai-bi)∙ui = 0

β is linear independent hence (ai - bi)= 0 ${\displaystyle \forall \!\,}$ i

i.e ai = bi, hence the combination is unique.

## Clarification on lecture notes

On page 3, we find that ${\displaystyle G\subseteq span(\beta )}$ then we say ${\displaystyle span(G)\subseteq span(\beta )}$. The reason is, the Theorem 1.5 in the textbook.

Theorem 1.5: The span of any subset ${\displaystyle S}$ of a vector space ${\displaystyle V}$ is a subspace of ${\displaystyle V}$. Moreover, any subspace of ${\displaystyle V}$ that contains ${\displaystyle S}$ must also contain ${\displaystyle span(S)}$

Since ${\displaystyle \beta }$ is a subset of ${\displaystyle V}$, ${\displaystyle span(\beta )}$ is a subspace of ${\displaystyle V}$ from the first part of the Theorem 1.5. We have shown (in the lecture notes page 3) that ${\displaystyle G\subseteq span(\beta )}$. From the "Moreover" part of Theorem 1.5, since ${\displaystyle span(\beta )}$ is a subspace of ${\displaystyle V}$ containing ${\displaystyle G}$, ${\displaystyle span(\beta )}$ must also contain ${\displaystyle span(G)}$.