When testing whether a variable is significant in a linear model, we perform the T test for the hypothese $H_0: \beta_j = 0$ vs $H_1: \beta_j \neq 0$:
$$ T = \frac{\beta_j}{\sqrt{\sigma^2[(X^TX)^{-1}]_{jj}}} $$
It's my understanding that when we interpret whether a variable is significant, based on the outcome of this test, it is in relation to the other variables being included in this model. Which part of this statistic takes the other variables into account? And is this test equivalent to performing an F-test between the model with $\beta_j$ and the model without it?