12-240/Classnotes for Tuesday October 2: Difference between revisions
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== Subspace == |
== Subspace == |
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Remind about the theorem of subspace: a subset W ⊂ V is a subspace iff is is closed under the operations of V |
Remind about the theorem of subspace: a non-empty subset W ⊂ V is a subspace iff is is closed under the operations of V and contains 0 of V |
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Proof: |
Proof: |
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First direction "=>": |
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if a non-empty subset W ⊂ V is a subspace , then W is a vector space over the operations of V . |
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=> + W is closed under the operations of V. |
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+ W has a unique identity of addition: <math>\forall\!\,</math> a <math>\in\!\,</math> W: 0 + a = a |
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Moreover, a a <math>\in\!\,</math> V. Hence 0 is also identity of addtition of V |
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Second direction "<=": |
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if a non-empty subset W ⊂ V is closed under the operations of V and contains 0 of V |
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we need to prove that W is a vector space over operations of V, hence, and subspace of V. |
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Namely, we need to show that W satisfies all axioms of a vector space, but now we just consider some axioms and leave the rest to readers. |
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VS1: Consider <math>\forall\!\,</math> x,y <math>\in\!\,</math> W => a,b <math>\in\!\,</math> V |
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While V is a vector space |
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thus x + y = y + x ( and the sum <math>\in\!\,</math> W since W is closed under addition) |
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VS2: (x + y) + z = x + (y + z) is proven similarly |
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VS3: As given, 0 of V <math>\in\!\,</math> W, pick any a in W ( possible since W is not empty) |
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So, a <math>\in\!\,</math> V hence a + 0 = a |
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Thus 0 is also additive identity element of W |
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== Class Notes == |
== Class Notes == |
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<gallery> |
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12-240-Oct-2-Page-1|Page 1 |
Image:12-240-Oct-2-Page-1.jpg|Page 1 |
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12-240-Oct-2-Page-2|Page 2 |
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12-240-Oct-2-Page-5|Page 5 |
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</gallery> |
</gallery> |
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Latest revision as of 04:43, 7 December 2012
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The "vitamins" slide we viewed today is here.
Today, the professor introduces more about subspace, linear combination, and related subjects.
Subspace
Remind about the theorem of subspace: a non-empty subset W ⊂ V is a subspace iff is is closed under the operations of V and contains 0 of V
Proof:
First direction "=>":
if a non-empty subset W ⊂ V is a subspace , then W is a vector space over the operations of V .
=> + W is closed under the operations of V.
+ W has a unique identity of addition: a W: 0 + a = a
Moreover, a a V. Hence 0 is also identity of addtition of V
Second direction "<=":
if a non-empty subset W ⊂ V is closed under the operations of V and contains 0 of V
we need to prove that W is a vector space over operations of V, hence, and subspace of V.
Namely, we need to show that W satisfies all axioms of a vector space, but now we just consider some axioms and leave the rest to readers.
VS1: Consider x,y W => a,b V
While V is a vector space
thus x + y = y + x ( and the sum W since W is closed under addition)
VS2: (x + y) + z = x + (y + z) is proven similarly
VS3: As given, 0 of V W, pick any a in W ( possible since W is not empty)
So, a V hence a + 0 = a
Thus 0 is also additive identity element of W
Class Notes
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Page 2
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Page 5
