I remember reading about beta and the GLM and how its calculated by using the matrix inverse, when I first started out reading about it. For some reason I formed the whole concept of individual correlations for each time point of a voxel and the subsequent distribution with its mean and sd as a way of representing what actually beta is representing, the weights of columns. Perhaps because the visualization came easier. Thankyou very much for taking time to point this out.
What I understand from your explanation is that contrast*beta is essentially the output of adding and subtracting the individual values of the betas, based on the 1's and the -1's in the contrast, 1 meaning the beta will be added and -1, it will be subtracted. I guess for my question about the direction of change in terms of betas, how does the t statistic differ between a certain value of contrast*beta and an equal but a negative value of the same. Since the absolute difference between the effect and the mean is the same, does it take into account whether the contrast*beta is a positive or a negative value?