Counting 1-D Subspaces of Z_3^3

[latentcorpse]

how many 1 dimensional subspaces of Z_3^3 are there?

Z_3^3 has 3^3 = 27 vectors

26 of which are non zero

then we can say v and 2v have the same span and so there are in fact 13 1 dimensional subspaces. is this true?

 

[dx]

No that doesn't make sense.

Hint: Draw the space as a lattice.

 

[latentcorpse]

for Z_3^2, my lecturer does the following:

there are 3^2=9 elements
8 are non zero
v and 2v have the same span
therefore there are 4 1 dimensional subspaces

why can't this be extended to Z_3^3

i don't know how to draw it as a lattice or what that means to be honest...

 

[dx]

Ok that does make sense, sorry. Here's why he argument works: The non-zero elements of the group Z₃ all have order three, so that means that if v is a non zero element of Z₃xZ₃, v and 2v will be distinct, and 3v will be (0, 0, 0), so {v, 2v, 3v} will be a one dimensional subgroup generated by v.