0

Is the words unordered bases and distinct bases of a vector space meaning same thing ?

Actually I have to solve the following problem.

$\underline{Problem}:$

Find the number of distinct bases of a two dimensional vector space of the finite field $\mathbb{F}_3$.

My approach:

Any two dimensional vector space $V$ over $\mathbb{F}_3$ can be considered as $V=\mathbb{F}_3^2$.

So we have to find the number of distinct bases (or unordered bases) of $\mathbb{F}_3^2$ over $\mathbb{F}_3$.

Now as $V$ is two dimensional any basis will be like $\{v_1,v_2 \}$.

So we have to check that given any non-zero vector $v_1 \in V=\mathbb{F}_3^2$, how many independent $v_2 \in V$ is there ?

Let $v_1$ is non-zero then $v_1$ has $9-1=8$ choices as there are $9$ elements in $\mathbb{F}_3^2$.

But how to count choices of $v_2$ for each choices of $v_1$ so that $\{v_1, v_2 \}$ are basis?

Help me

Community
  • 1
MAS
  • 10,898
  • 2
    Possible duplicate of Number of bases of an n-dimensional vector space over q-element field.. It is possible that better duplicates exist. I simply picked the top choice from the list of related posts in the right margin - no need to search anything. – Jyrki Lahtonen Aug 18 '19 at 19:57
  • This question has been asked many times. Often leading to a formula for the number of invertible $n\times n$ matrices over a finite field – Jyrki Lahtonen Aug 18 '19 at 19:58
  • 5
    No, in this question the word “distinct” is not a direct modifier of “basis”; it is just part of the expression “number of distinct”. You could equally ask for the number of distinct unordered bases or the number of distinct ordered bases. It’s a matter of judgment to interpret this to mean “unordered”: the word “distinct” mildly suggests this, only because it begs the question “distinct up to what?”, and “up to ordering” is one reasonable interpretation. – Erick Wong Aug 18 '19 at 20:07
  • @ErickWong, I did not get you. My question is- whether the unordered bases and distinct bases meaning same thing or not. According to your comment, I can say that these are not same term. is it ? – MAS Aug 18 '19 at 20:55
  • 3
    @M.A.SARKAR Correct. Not only are they not the same term, they are not even comparable. There is no such term as “distinct basis”. Here the question is asking about “the number of distinct X” where X is a basis. – Erick Wong Aug 18 '19 at 20:57
  • @ErickWong, thank you. I now know there are duplicate of the 2nd part of my question. But I did an approach , chosen $v_1$ but got stuck to choose $v_2$. Some information says that it is $3^2-3=6$. But why? Can you explain it please? – MAS Aug 18 '19 at 20:58
  • 1
    @M.A.SARKAR That should be answered in the duplicate. There are exactly $3$ vectors (including $0$) that are scalar multiples of $v_1$. – Erick Wong Aug 18 '19 at 21:18
  • @ErickWong, ok, I got it. Thanks – MAS Aug 18 '19 at 21:19

1 Answers1

1

Converting comments into a full-fledged answer:

No, these adjectives don’t mean the same thing. In fact, it doesn’t even make sense to compare them directly, because “distinct basis” does not mean anything. When the question asks to count the number of distinct bases, it does not mean “count the number of X such that X is a distinct basis”, which is a meaningless concept. Instead, it simply means “count the number of distinct X such that X is a basis”. In this regard, the word “distinct” is somewhat redundant, as we generally assume that we are counting distinct instances whenever counting any object.

So the question itself does not clearly specify whether it is asking for ordered bases or unordered bases. It could easily mean either, depending on the context: some books use “basis” to mean ordered and some mean unordered. I suspect many books state a default convention for what an unqualified “basis” means, and uses the opposite term explicitly when needed.

In the absence of a stated convention, there is a connotation here that the question does suggest “unordered”. The reason is that they go to the trouble of using the word “distinct” even though that is implicit in counting any object. So this begs the question: “distinct up to what?”. In what possible sense could two ordered bases could be considered the same? The answer that springs to mind is that they might be the same up to ordering, in which case counting the number of distinct bases (up to reordering ordered bases) would be equivalent to counting unordered bases.

Erick Wong
  • 25,868
  • 3
  • 43
  • 96