12-240/Classnotes for Tuesday October 09

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In this lecture, the professor concentrate on basics and related theorems.

Definition of basic

β V is a basic if

1/ It generates ( span) V, span β = V

2/ It is linearly independent

theorems

1/ β is a basic 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 ai.ui = 0 ai F, ui β

ai.ui = 0 = 0.ui

since 0 can be written as a linear combination of elements of β in a unique way, ai=0 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.

Lecture notes scanned by Oguzhancan