09-240/Classnotes for Tuesday October 20

Def

V & W are "isomorphic" if there exists a linear transformation T:V → W & S:W → V such that T∘S=I_W and S∘T=I_V

Theorem

If V& W are field dimensions over F, then V is isomorphic to W iff dim V=dim W

Corollary

If dim V = n then $\mathrm {V} \cong \mathrm {F^{n}}$

Note: $\cong$ represents isomorphism

Two "mathematical structures" are "isomorphic" if there's a "bijection" between their elements which preserves all relevant relations between such elements.

Example: Plastic chess is "isomorphic" to ivory chess, but it is not isomorphic to checkers.

Ex: The game of 15. Players alternate drawing one card each. Goal: To have exactly three of your cards add to 15.

O: 7, 4, 6, 5 → Wins! X: 3, 8, 1, 2

This game is isomorphic to Tic Tac Toe!

4	9	2
3	5	7
8	1	6

Converts to:

O	9	X
X	O	O
X	X	O