Predict() new data into PCA space in R - r

Question

After performing a principal component analysis of a first data set (a), I projected a second data set (b) into PCA space of the first data set.
From this, I want to extract the variable loadings for the projected analysis of (b). Variable loadings of the PCA of (a) are returned by prcomp(). How can I retrieve the variable loadings of (b), projected into PCA space of (a)?
# set seed and define variables
set.seed(1)
a = replicate(10, rnorm(10))
b = replicate (10, rnorm(10))
# pca of data A and project B into PCA space of A
pca.a = prcomp(a)
project.b = predict(pca.a, b)
# variable loadings
loads.a = pca.a$rotation

Answer 1

Here's an annotated version of your code to make it clear what is happening at each step. First, the original PCA is performed on matrix a:
pca.a = prcomp(a)
This calculates the loadings for each principal component (PC). At the next step, these loadings together with a new data set, b, are used to calculate PC scores:
project.b = predict(pca.a, b)
So, the loadings are the same, but the PC scores are different. If we look at project.b, we see that each column corresponds to a PC:
PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8
[1,] -0.2922447 0.10253581 0.55873366 1.3168437 1.93686163 0.998935945 2.14832483 -1.43922296
[2,] 0.1855480 -0.97631967 -0.06419207 0.6375200 -1.63994127 0.110028191 -0.27612541 -0.37640710
[3,] -1.5924242 0.31368878 -0.63199409 -0.2535251 0.59116005 0.214116915 1.20873962 -0.64494388
[4,] 1.2117977 0.29213928 1.53928110 -0.7755299 0.16586295 0.030802395 0.63225374 -1.72053189
[5,] 0.5637298 0.13836395 -1.41236348 0.2931681 -0.64187233 1.035226594 0.67933996 -1.05234872
[6,] 0.2874210 1.18573157 0.04358772 -1.1941734 -0.04399808 -0.113752847 -0.33507195 -1.34592414
[7,] 0.5629731 -1.02835365 0.36218131 1.4117908 -0.96923175 -1.213684882 0.02221423 1.14483112
[8,] 1.2854406 0.09373952 -1.46038333 0.6885674 0.39455369 0.756654205 1.97699073 -1.17281174
[9,] 0.8573656 0.07810452 -0.06576772 -0.5200661 0.22985518 0.007571489 2.29289637 -0.79979214
[10,] 0.1650144 -0.50060018 -0.14882996 0.2065622 2.79581428 0.813803739 0.71632238 0.09845912
PC9 PC10
[1,] -0.19795112 0.7914249
[2,] 1.09531789 0.4595785
[3,] -1.50564724 0.2509829
[4,] 0.05073079 0.6066653
[5,] -1.62126318 0.1959087
[6,] 0.14899277 2.9140809
[7,] 1.81473300 0.0617095
[8,] 1.47422298 0.6670124
[9,] -0.53998583 0.7051178
[10,] 0.80919039 1.5207123
Hopefully, that makes sense, but I'm yet to finish my first coffee of the day, so no guarantees.