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Vectors Cheat Sheet (DRAFT) by

Vector operations and stuff

This is a draft cheat sheet. It is a work in progress and is not finished yet.

Axioms For Vector Spaces

To show that a given set with two operations is NOT a vector space, we need to show properly that at least one property of the ten above is violated.
(1) If u, v∈V, then u⊕v∈V [CLOSURE UNDER ADDITION]
(2) If u⊕v=v⊕u [COMMM­UTATIVE LAW]
(3) If (u⊕v)⊕w=u⊕(v⊕u) ['⊕' IS ASSOCI­ATIVE]
(4) V contains the object "­0" which satisfies u⊕0=0⊕u
For each u∈V, there exist an object '-u' such that u⊕-u=0 [ADDITIVE INVERSE]
(6) If u∈V*and k∈K, then k⊙u∈V [CLOSURE UNDER MULTIP­LIC­ATION]
(7) k⊙(u⊕v)=(k⊙u)⊕(k⊙v) [DISTR­IBUTIVE LAW]
(8) (k+l)⊙u=(k⊙u)⊕(l⊙u)
(9) k⊙(l⊙u)=(kl⊙u)
(10) 1⊙u=u

Axioms For Vector Spaces

If u, v∈V, then u⊕v∈V [CLOSURE UNDER ADDITION]
 
 
       
 
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