CUNY DATA 605 - Week 1 Homework

Problem set 1

(1) Calculate the dot product u:v where u = [0.5; 0.5] and v = [3;4]

u <- c(0.5, 0.5)
v <- c(3, 4)

dot.uv <- sum(u*v)
dot.uv

## [1] 3.5

Could also have used:
* t(u) %% v
crossprod(u,v)

(2) What are the lengths of u and v? Please note that the mathematical notion of the length of a vector is not the same as a computer science definition.

norm.u <- sqrt(sum(u*u))
norm.u

## [1] 0.7071068

Could also have used:
* norm(u, type=“2”)

norm.v <- sqrt(sum(v*v))
norm.v

## [1] 5

Could also have used:
* norm(v, type=“2”)

(3) What is the linear combination: 3u +􀀀 2v?

3 * u + 2 * v

## [1] 7.5 9.5

(4) What is the angle between u and v?

Making use of the formula:

rad <- acos(dot.uv / (norm.u * norm.v))
print(paste0("The angle between the vectors is: ", round(rad * 180/pi,3), " degrees"))

## [1] "The angle between the vectors is: 8.13 degrees"

Problem set 2

Set up a system of equations with 3 variables and 3 constraints and solve for x. Please write a function in R that will take two variables (matrix A & constraint vector b) and solve using elimination. Your function should produce the right answer for the system of equations for any 3-variable, 3-equation system. You don’t have to worry about degenerate cases and can safely assume that the function will only be tested with a system of equations that has a solution.

…The approach that you should employ is to construct an Upper Triangular Matrix and then back-substitute to get the solution…

First, I create a function to decompose the A matrix, verifying at the end that AX = B.

A <- NA # initialize matrices
U <- NA
L <- NA

decomp <- function(mat){
  A <- matrix(mat, nrow = 3, byrow = T) # construct matrices from 9-dim vector
  print("Matrix A")
  print(A)
  print("")
  
  U <- A
  
  a1 <- U[2,1]/U[1,1] # decompose Upper Matrix
  U[2,] <- U[1,] * -(a1) + U[2,]
  
  a2 <- U[3,1]/U[1,1]
  U[3,] <- U[1,] * -(a2) + U[3,]
  
  a3 <- U[3,2]/U[2,2]
  U[3,] <- U[2,] * -(a3) + U[3,]
  
  print("Matrix U")
  print(U)
  print("")
  
  L <- matrix(0, nrow = 3, ncol = 3, byrow = T) # initialize Lower Matrix
  L[1,1] <- 1; L[2,2] <- 1; L[3,3] <- 1
  L[2,1] <- a1 # decompose Lower Matrix
  L[3,1] <- a2
  L[3,2] <- a3
  
  print("Matrix L")
  print(L)
  print("")
  
  LU <- L %*% U ## check that A = LU
  print("Matrix Decomposition LU")
  print(LU)
  print("")
  
  print(paste0("The original matrix and multiplication of decomposed LU matrices is the same: ", identical(A,round(LU,4))))
  
  U <<- U ## save as global variables to retain for solve function
  L <<- L
}

Next, I create a second function to imput the constraint vector.

## Recall AX = B, since A = LU
## LUX = B
## solve LY = B for Y, then
## UX = Y for X

Y <- NA # initialize factors
X <- NA

solve.eq <- function(constraint){
  B <- constraint
  Y[1] <- B[1]  # perform backward substitution
  Y[2] <- B[2] - Y[1] * L[2,1]
  Y[3] <- B[3] - Y[1] * L[3,1] - Y[2] * L[3,2]
  
  X[3] <- Y[3] / U[3,3]
  X[2] <- (Y[2] - X[3] * U[2,3])/U[2,2] 
  X[1] <- (Y[1] - X[2] * U[1,2] - X[3] * U[1,3])/U[1,1]
  print(round(X,2))
}

Finally, I solve with the test matrices provided.

decomp(c(1,1,3,2,-1,5,-1,-2,4))

## [1] "Matrix A"
##      [,1] [,2] [,3]
## [1,]    1    1    3
## [2,]    2   -1    5
## [3,]   -1   -2    4
## [1] ""
## [1] "Matrix U"
##      [,1] [,2]      [,3]
## [1,]    1    1  3.000000
## [2,]    0   -3 -1.000000
## [3,]    0    0  7.333333
## [1] ""
## [1] "Matrix L"
##      [,1]      [,2] [,3]
## [1,]    1 0.0000000    0
## [2,]    2 1.0000000    0
## [3,]   -1 0.3333333    1
## [1] ""
## [1] "Matrix Decomposition LU"
##      [,1] [,2] [,3]
## [1,]    1    1    3
## [2,]    2   -1    5
## [3,]   -1   -2    4
## [1] ""
## [1] "The original matrix and multiplication of decomposed LU matrices is the same: TRUE"

solve.eq(c(1,2,6))

## [1] -1.55 -0.32  0.95

The result is confirmed.