Linearity between predictors and dependent variable in a linear model I run the following linear model in R :
lm(formula = NA. ~ PC + I(1/SPCI), data = DSET)

The p-value for each predictor is significant, and it works fairly well with respect to most of the assumptions in linear regression, such as:


*

*Normal distribution of errors.

*High correlation between predicted values and estimated values.

*Homoscedasticity.

*Non-collinearity between the predictors: PC and (1/SPCI) are not correlated at all!


But, digging more into the topic, there's an assumption that fails in my model, and it says: 

Linearity of the relationship between dependent and independent variables. 

This kind of contradicts the non-collinearity assumption because, if NA. and PC are highly correlated and NA. and (1/SPCI) are too, then PC and (1/SPCI) are correlated, and this is violating the assumption of non-collinearity. 
I think that I misunderstood the assumption, or there's an explanation to this.
 A: The emphasis upon so called assumptions in linear regression modeling is evidence that  pedagogy in the applied fields is not in line with traditional statistical theory. In particular, the aforementioned "straight line assumption" (or non-existence of higher order effects) is completely erroneous depending upon your intended application. 
For inference, the test of whether a regression parameter $\beta = 0$ under the null hypothesis, it should be stressed that the $\beta$ need not model the truth, it's just a first order trend indicating the direction (sign) and strength (value) of that association. Such is the case with regression models and the name itself: a simplified "rule of thumb". Smoking is negatively associated with survival, social drinking is positively associated with obesity, etc. are examples of such rules of thumb gleaned from rules-of-thumb regression models.
For prediction, the complexity of a trend can be infinite provided a sufficient quantity of data have been provided for the application. Thus it's moot to ask "is a straight line enough" when we should be asking, "would a fractional polynomial / smoothing spline / etc. do better?" that decision, of course, would be based upon the overall predictive accuracy determined in a split-sample cross-validated independent dataset. 
In both cases, making decisions based on visual inspections about the types of models you fit greatly increase the risk of overfitting the trend (committing type I errors in the inference case). I think the answer is to consider exactly what the regression model is intended to do and determine how flexible and robust the model is for that application.
