Why I am getting different coefficients for R's classification model?

Question

Dataset

Consider the following dataset which measures the price of a computer given different configurations

Situation

After applying the four-way classification model as below, we have the summary output of the model and the relevant coefficients:

fullwithinteract <- lm(Price ~CPU + Speed + Floppy + Monitor + CPU:Speed + CPU:Floppy + CPU:Monitor + Speed:Floppy + Floppy:Monitor,Q10set1)

Problem

The problem arises when I tried to reproduce the coefficients output by R using a design matrix. I first consider the following design matrix:

And I find the coefficients by the normal equation

solve(t(X)%*%X) %*% t(X) %*% Y

where $X$ is the design matrix above and $Y$ is the Price response vector.

The output is        
V1   4688.9286
V2  -2872.5000
V3  -1388.9286
V4   2326.0714
V5    482.5000
V6   1252.2619
V7   -742.5000
V8    837.7381
V9  -2126.0714
V10    17.5000

which is drastically different from the coefficients output by R, EXCEPT in some entries. Therefore, how should I interpret the coefficients output by R? Is R calculating the coefficients in the same way as I did using the normal equation? Should I even use the normal equation to calculate the coefficients?

Data availability

All the data and the design matrix used in the above presentation can be obtained in the following website: https://www.notion.so/hephaes/Data-Coefficients-by-R-e161d467a09948028a43633aaf616229

Answer 1

You can get the design matrix from R formula

Here is an example.

> model.matrix(mpg~wt+wt*cyl,mtcars)
                    (Intercept)    wt cyl wt:cyl
Mazda RX4                     1 2.620   6 15.720
Mazda RX4 Wag                 1 2.875   6 17.250
Datsun 710                    1 2.320   4  9.280
Hornet 4 Drive                1 3.215   6 19.290
Hornet Sportabout             1 3.440   8 27.520
Valiant                       1 3.460   6 20.760

Try to set

X = model.matrix(Price ~CPU + Speed + Floppy + Monitor + CPU:Speed + CPU:Floppy + CPU:Monitor + Speed:Floppy + Floppy:Monitor,Q10set1)

Then use normal equation to solve.