12-240/Classnotes for Tuesday October 09

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

Definition of basis

β 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 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.

So, suppose ai∙ui = v = bi∙ui

Thus ai∙ui - bi∙ui = 0

(ai-bi)∙ui = 0

β is linear independent hence (ai - bi)= 0 i

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

Clarification on lecture notes

On page 3, we find that then we say . The reason is, the Theorem 1.5 in the textbook.

Theorem 1.5: The span of any subset of a vector space is a subspace of . Moreover, any subspace of that contains must also contain

Since is a subset of , is a subspace of from the first part of the Theorem 1.5. We have shown (in the lecture notes page 3) that . From the "Moreover" part of Theorem 1.5, since is a subspace of containing , must also contain .

Lecture notes scanned by Oguzhancan

Lecture notes uploaded by gracez