Which one of the following statements is ALWAYS correct about a collection of p column vectors, each having n real-valued entries?

The correct option is: If p > n, then the column vectors must be linearly dependent

Let us break down the reasoning behind this statement step-by-step to understand why it is always correct:
First, we are given a collection of $p$ column vectors, where each vector has $n$ real-valued entries. This means each of these vectors belongs to the vector space ${ℝ}^n$.

The dimension of the vector space ${ℝ}^n$ is exactly $n$. By definition, the dimension of a vector space is the maximum number of linearly independent vectors that can exist in that space.

Therefore, any set containing more than $n$ vectors in ${ℝ}^n$ must be linearly dependent. Since $p$ represents the number of vectors, if $p>n$, the number of vectors exceeds the dimension of the space, making it mathematically impossible for all of them to be linearly independent.
Consequently, they must be linearly dependent.