R: Least squares estimation in linear equation systems

Question

I'm going to estimate some parameters of linear equation systems with repeated measures. My equations will look like this:

Variant1:

Variant2:

At least 10 values (repeated measures; technical replicates) are known for every and . I want to estimate the values for and resp. .

Additionally I'd like to know the standard error of these estimates, if possible.

In R, my data set would look like this (in reality I have :

i <- rep(1:3, each = 30)
j <- rep(rep(1:3, each = 10), 3)

K.i  <- rep(c(6, 5, 10), each = 30) + rnorm(90)
K.ij <- K.i + rnorm(90)

# X_i, X_ij and x_ij should be 0 (since I assumed K_j being K_ij + normal noise

data <- cbind(i, j, K.i, K.ij)

How can I estimate the expected parameter values (minimizing the sums of squares) and the standard errors of these estimates in R?

Thanks a lot in advance for your help!