Are vectors generated from the same normal distribution linearly independent?

Question

I have a normal distribution with zero mean and unit variance. I then generated a 100 dimensional vector v1, each element number is from the distribution.

Now I generated another 9 vectors: v2, v3, ... v10 using the same approach. Then are these 10 vectors linearly independent?

Also, if I put them together as a 100x10 matrix: [ v1, v2, ... v10], what would be the rank?

Thanks!

Answer 1

Based on the wording in your question, I assume each entry is generated i.i.d. from $N(0,1)$. This means each vector $v_i$ is a standard Gaussian vector from the distribution $N(0, I_{100})$, and the $v_i$ are [probabilistically] independent. In particular, the probability that $v_i$ lies in any strict subspace of $\mathbb{R}^{100}$ is zero, since strict subspaces have no interior (no "volume").

Thus the probability of $v_2$ lying in the span of $v_1$ (given $v_1$) is zero, so $\{v_1, v_2\}$ is linearly independent with probability $1$.

Similarly, the probability of $v_3$ lying in the span of $\{v_1, v_2\}$ is also zero, so $\{v_1, v_2, v_3\}$ is linearly independent with probability $1$.

Continuing in this fashion, you can show that your $100 \times 10$ matrix has rank $10$ with probability $1$.