In the vector space C3, compute the vector representation \(p_B(v)\) for the basis B and vector v below:
# B is a collection of three basis vectors
B <- cbind(c(2,-2,2), c(1,3,1), c(3,5,2))
B## [,1] [,2] [,3]
## [1,] 2 1 3
## [2,] -2 3 5
## [3,] 2 1 2# v is a column vector
v <- rbind(11, 5, 8)
v## [,1]
## [1,] 11
## [2,] 5
## [3,] 8# to linearly transform v by B
v_trans <- B %*% v
v_trans## [,1]
## [1,] 51
## [2,] 33
## [3,] 43# and just to double check, transforming v_trans by B^-1 should get us back to v
solve(B) %*% v_trans## [,1]
## [1,] 11
## [2,] 5
## [3,] 8