Again, I am going to use R to solve this problem. N.B., that in grad school for physics we had to solve a lot of problems like this by hand, on pencil and paper. Effectively what I hope to get from this class is to learn new ways of letting the software do the heavy lifting, which is something old-school Physics Profs (Manhattan Project era) found to be “trivial.” They still had shoe boxes full of punch cards on thier book shelves.
I have chosen c.17, in the Vector spaces chapter.
Working within the vector space C4, determine if b = [1,1,0,1] is in the subspace W,
W = < { [1,2,-1,1], [1,0,3,1], [2,1,1,2] } >
At the heart of this problem is makeing an Augmented matrix [W|b] then finding the reduced row echalon form of the Augmented matrix. Someone has written a function to do that in the package “pracma” (http://www4.ncsu.edu/~slrace/LAprimer/LAWorksheets/Row_Reduction_in_R.pdf)
Note that rref has a dependency on package “quadprog”, so you have to install that first.
#install.packages("quadprog", dependencies = TRUE)
#install.packages("pracma", dependencies = TRUE)
library("pracma")## Warning: package 'pracma' was built under R version 3.4.1Now to make the Augmented Matrix
c17 <- matrix(c(1,2,-1,1,1,0,3,1,2,1,1,2,1,1,0,1), nrow = 4, ncol= 4)
c17## [,1] [,2] [,3] [,4]
## [1,] 1 1 2 1
## [2,] 2 0 1 1
## [3,] -1 3 1 0
## [4,] 1 1 2 1We now have to find the reduced row echalon form, for this we will use the rref function in pracma.
rref(c17)## [,1] [,2] [,3] [,4]
## [1,] 1 0 0 0.3333333
## [2,] 0 1 0 0.0000000
## [3,] 0 0 1 0.3333333
## [4,] 0 0 0 0.0000000Note that the final column is not a pivot column, so the system is consistant and b in the subspace W.