Vector Space Span and Linear Independence in Z_2 Field

[JG89]

Homework Statement

Let $$S = \{ u_1, u_2, ... , u_n \}$$ be a linearly independent subset of a vector space V over the field $$Z_2$$. How many vectors are there in span(S)?

Homework Equations

The Attempt at a Solution

I don't know what the field $$Z_2$$ is. It's not explained in the text either. Any ideas?

 

[JG89]

Could it mean the field of integers, addition and multiplication modulo 2?

 

[Mark44]

It's the integers modulo 2. All of the integers are mapped to 0 or 1, depending on whether the remainder in division by 2 is 0 or 1. Z₂ is an example of a finite field. The set of linear combinations of the vectors in S is therefore finite.

 

[JG89]

Possible linear combinations of S:

0*u_1 + 0 *u_2 + ... + 0*u_n
1*u_1 + 0*u_2 + 0*u_3 ... + 0*u_n
1*u_1 + 1*u_2 +0*u_3 + ... + 0* u_n
.
.
.
.
.

And so on. Is there a fast way to do this?

 

[Mark44]

Yes. Work up to it. If your set S has one vector in it, how many vectors would be in span(S)? Answer: 2
If S has two vectors, how many vectors in span(S).
If S has three vectors, how many vectors in span(S).
You should soon see a pattern.

 

[JG89]

Ah, I see. I've wrote out possible linear combinations, and it seems like span(S) will have 2^n vectors.

Thanks Mark!

 

[JG89]

For the question, should I write out a formal proof for this? Or is it sufficient enough to write out the linear combination's of a set with one vector, then the linear combination's of a set with two vectors, then the linear combination's of a set with three vectors, then just say that the pattern is 2^n where n is the amount of vectors.

 

[Mark44]

That would probably work, but if you really wanted to be rigorous about it, you could prove it by math induction, which would be pretty simple in this problem.

OTOH, if all you need to do is to answer the question, not prove it, a simple explanation would be fine. You might want to check with your prof to see what he/she is looking for.