“be n-dimensional vector of variables” vs “be an n-dimensional vector of variables”

Question

For the determiner in English, I learn that they are used in many different situations. For example, a is used before a noun. However, sometimes I misuse it. Therefore, I would like to understand it, so I can write my sentence correctly.

For example,

I have a vector of random variables.

As I understand, a in this sentence is a determiner used to identify the vector, not the random variables. Hence, it is correct to say:

Let X be a vector of random variables.

Because in this sentence, the determiner a refers to the vector X and did not refer to the random variables. Is that correct? if not, then, why?

However, similar to this sentence, I would like to specify the dimension of this vector. So, is it ok to say:

Let X be an n-dimensional vector of random variables.

Or I must not use the determiner an here because n-dimensional is plural in its meaning. Is that correct?

So, which sentence is correct?

Answer 1

Yes, that's correct.

The essential noun in the sentence is vector, which requires an indefinite article:

I have a vector.

This doesn't change if you qualify it with an additional noun phrase, no matter if that additional portion of the noun phrase is singular or plural:

I have a vector (of X and Y).

Nor does it change if you use an adjective in front of it, regardless of the singularity or plurality of the adjective:

I have a (Z) vector (of X and Y).

The consideration of the article is based only on the noun itself, not how it's being modified by other things.

---

I say that, but in the one example you gave, the form of the indefinite article changes from a to an because of the vowel sound of n-dimensional that comes immediately after it:

Let X be an n-dimensional vector of random variables.

But if the adjective started with a consonant sound, you would use a again:

Let X be a four-dimensional vector of random variables.

Both a and an are used for a singular noun: vector in this case.

---

If vector itself became plural, then the indefinite article would be dropped because you would no longer have a single vector:

I have vectors.
I have vectors of random variables.
I have vectors that are long.
I have single-dimensional vectors of random variables.
I have n-dimensional vectors of random variables.

The consideration of using the indefinite article or not is based only on the use of vector or vectors.