Find the cardinality and dimension of the vector space [itex]\mathbb{Z}^{3}_{7}[/itex] over [itex]\mathbb{Z}_{7}[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

[itex]\mathbb{Z}^{3}_{7} = \{ (a,b,c) \; | \; a,b,c \in \mathbb{Z}_{7} \}[/itex].

Then since [itex]\mathbb{Z}_{7}[/itex] is a field [itex]1 \cdot a = a \; \forall \; a[/itex], so [itex]B = \{ (1,0,0), (0,1,0) , (0,0,1) \}[/itex] is a basis of [itex]\mathbb{Z}^{3}_{7}[/itex], so [itex]\dim \mathbb{Z}^{3}_{7} = 3[/itex]. ans = 9, what the?

Thanks

James

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Cardinality and dimension

**Physics Forums | Science Articles, Homework Help, Discussion**