Final answer:
A variable's dimension after PCA indicates its influence on the direction of maximum variance. High loadings on a principal component reflect strong influences.
Step-by-step explanation:
The dimension of a variable after Principal Component Analysis (PCA) tells us about the variable's contribution to the direction of maximum variance in the data set. PCA is performed to reduce the dimensionality of large data sets, transforming the data into a new set of dimensions called principal components, which are orthogonal (uncorrelated) to each other. Each principal component represents a different amount of the dataset's total variance, with the first principal component accounting for the largest portion. The coefficients in the principal component, also known as "loadings," quantify the relative contribution of each original variable to this principal axis. Therefore, a high loading of a variable on a particular principal component indicates that it has a strong influence on the direction along which the data varies the most along that component.