12-240/Classnotes for Tuesday October 2

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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 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: \forall\!\, a \in\!\, W: 0 + a = a

Moreover, a a \in\!\, 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

we need to prove that

Class Notes