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Week of...
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Notes and Links
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1
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Sep 10
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About This Class, Tuesday, Thursday
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2
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Sep 17
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HW1, Tuesday, Thursday, HW1 Solutions
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3
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Sep 24
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HW2, Tuesday, Class Photo, Thursday
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4
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Oct 1
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HW3, Tuesday, Thursday
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5
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Oct 8
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HW4, Tuesday, Thursday
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6
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Oct 15
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Tuesday, Thursday
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7
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Oct 22
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HW5, Tuesday, Term Test was on Thursday. HW5 Solutions
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8
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Oct 29
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Why LinAlg?, HW6, Tuesday, Thursday, Nov 4 is the last day to drop this class
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9
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Nov 5
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Tuesday, Thursday
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10
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Nov 12
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Monday-Tuesday is UofT November break, HW7, Thursday
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11
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Nov 19
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HW8, Tuesday,Thursday
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12
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Nov 26
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HW9, Tuesday , Thursday
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13
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Dec 3
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Tuesday UofT Fall Semester ends Wednesday
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F
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Dec 10
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The Final Exam (time, place, style, office hours times)
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Register of Good Deeds
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Add your name / see who's in!
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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