12-240/Classnotes for Tuesday October 2: Difference between revisions
From Drorbn
Jump to navigationJump to search
m (→Subspace) |
|||
| Line 8: | Line 8: | ||
== Subspace == |
== 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 |
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 contain 0 of V |
||
Proof: |
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: <math>\forall\!\,</math> a <math>\in\!\,</math> W: 0 + a = a |
|||
Second direction |
|||
if a non-empty subset W ⊂ V is closed under the operations of V |
|||
we need to prove that |
|||
== Class Notes == |
== Class Notes == |
||
Revision as of 11:56, 4 October 2012
| |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 contain 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
Second direction
if a non-empty subset W ⊂ V is closed under the operations of V
we need to prove that
Class Notes
Page 1
Page 2
Page 3
Page 4
Page 5
