Question

what does \[(\mathbb{Z}/7\mathbb{Z})^2\] mean?

Answer 1

hmm...i dont think ive seen that notation before =/ if i were to make a guess, its that you are squaring all the elements of:
\[\mathbb Z/7\mathbb Z\]
but i dont know for sure. Is it in relation to some problem? maybe if you post the problem we can get some context clues?

Answer 2

In the vector space \[(\mathbb{Z}/7\mathbb{Z})^4\]determine whether the set \[\left\{ \left( 1,3,0,2 \right), \left( 2,1,3,0 \right) \right\}\]is linearly dependent and whether it is a basis.

Answer 3

Well it cant be a basis for sure. Its a 4 dimensional space, and you only have 2 vectors. So we just have to make sure they are linearly independent.

Answer 4

er, i guess i should rephrase that, we need to check if they are linearly independent or not.

Answer 5

oh but it is definitely not a basis? why though? is the 4 dimensional space based on the number of elements in a single vector?

Answer 6

yes.

Answer 7

oh all right. and to find out wether it is linearly independent, i have to carry out the row-reduce-echelon form yes?

Answer 8

i want to say yes. but to be honest, ive never carried out row reduction when the elements in the matrix are a part of:
\[\mathbb Z/m\mathbb Z\]
I dont see why it wouldnt work though.

Answer 9

I believe they are independent. looking at the 4th element in each vector, there is no scalar multiple of 2 that will be congruent to 0 modulo 7. (or vice-versa)

Answer 10

haha all right. yeah. i am giving it a shot. just going to try and reduce it in the normal \[\mathbb{Z}\]form and not the \[(\mathbb{Z}/7\mathbb{Z})\]form first. Just to confirm though, the vectors are linearly independent if the number of non-zero rows = number of vectors yes?

Answer 11

They will be linearly independent if you end up with this:

1 0
0 1
0 0
0 0

assuming you make the vectors into columns.

Answer 12

yes. that's what i got. thanks heaps (:

and also. does span mean what you said just now about the 2 vectors and the 4 dimensional vector space?

Answer 13

yes. the questions, "Do this set of vectors form a basis for the space?" and "Does this set of vectors span the space" mean the same thing. Since its a 4 dimensional space, you would need 4 vectors to span it.

Answer 14

er...4 linearly independent vectors.

Answer 15

oh right. makes more sense now. thanks a lot! (: