Homework 2 Questions

Problem 1 (Golan 56)

Question. I'm not quite sure what is being asked. Z/(4)={0,1,2,3} with addition modulo 4. Similarly, GF(2)=Z/(2)={0,1} with addition modulo 2. Are we being asked to come up with a clever way of writing addition on Z/(4) so that no matter what we add, we get elements in Z/(2) (i.e. 0 or 1)?

Answer. Good question. No, the four elements {0,1,2,3} of Z/(4) are the vectors in the desired vector space. The two elements {0,1} are the scalars in the field.

This problem can be confusing if you don't have experience with these things. The confusion arises because there are two 0's and two 1's floating around, and it's important to keep them distinct. It helps to give a special label to the field elements. For example, we could call them 0_F and 1_F.

Now, you are told that vector addition should be the usual addition mod 4 on Z/(4). So, you just need to come up with some definition for scalar multiplication so that the vector space rules are satisfied. That is, you must define "0_F times x" for each x in {0, 1, 2, 3}. Similarly, you must define "1_F times x" for each x in {0, 1, 2, 3}. Then check that the resulting algebraic structure --- that is, (Z/(4), +, -, 0, 0_F times, 1_F times) --- is a vector space.

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Problem 7 (Golan 81)

There is a typo. The first instance of the word "subspace" should be replaced with the word "subset."