Background.
The multiple linear regression model is of the form
$$ Y = \beta_0 + \beta_1X_1 + \cdots + \beta_nX_n + \epsilon $$
where we assume $\epsilon$ is normally distributed with constant variance. One way to decide whether any of the predictors are useful in predicting the response $Y$ is to test the following null hypothesis against the alternative:
$$ H_0: \beta_1 = \cdots = \beta_n =0 $$
versus
$$ H_1: \beta_j \neq 0, \text{for at least one } j = 1, \ldots,n $$
To test this, we can use the $F$ statistic.
Question.
Rather than using the $F$ statistic, since $X_1, \ldots, X_n$ are observed and are vectors, would it make any sense to test $H_0$ by checking whether $\{X_1, \ldots, X_n\}$ is a linearly independent set? What I mean is, checking whether the only solution to the equation
$$ \beta_1X_1 + \cdots + \beta_nX_n = 0 $$
is $\beta_1 = \cdots = \beta_n =0$.
If no, is it because we're not taking into account $\epsilon$?
Sanity check: Your proposal does not involve $Y$ in any way, which suggests it doesn't really address the root question.
More detail: For most datasets, $X_1,\ldots, X_n$ are linearly independent (and in linear regression, we actually prefer this to avoid collinearity issues). And in many of those cases, some subset of them may be a good predictor of $Y$. So linear independence of $X_1, \ldots, X_n$ does not really correspond to $H_0$ here.