Linear regression rests on a few assumptions about the data and the residuals. The relationship between the features and the target is linear. Observations are independent of each other. Residuals have constant variance across the fitted range. Residuals are roughly normal. And no feature is a perfect linear combination of the others.
They matter in two different ways. Linearity and independence affect the coefficients themselves, so breaking them makes the estimates wrong. Constant variance and normal residuals mostly affect the uncertainty around those estimates. The fitted line can look fine while every p-value and confidence interval quietly lies.
The cheapest check is a residual plot against the fitted values. A funnel shape means the variance grows with the prediction. A curve means you are forcing a straight line through a bend. Time-ordered or repeated-measure data usually breaks independence. That makes the model look far more confident than it deserves.
Rewriting in plainer words…
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