Problem set #1

  1. u.v = (0.5)3 + (0.5)-4 = 1.5 - 2 = -0.5

  2. the lenghts of both vectors are 2x1

  3. 3u - 2v = [1.5; 1.5] - [6;-8] = [-4.5; 9.5]

  4. |u| = sqrt(0.5^2 + 0.5^2) = 0.7071 |v| = sqrt(3^2 + (-4)^2) = 5

using the formula θ = arccos (u.v / |u||v|) = arccos (-0.5/0.7071*5) = 1.0100168

approximately 57.86 degrees.

Problem set #2

gauss = function(A, b){
  r <- dim(A)[1]
  c <- dim(A)[2] + dim(b)[2] # full size
  full <- matrix(c(A, b), nrow=r, ncol=c)
  
  # construct a upper triangular matrix,with that, the last row will be reduced.
  for (j in 1:(c-2)) {
    for (i in (j+1):r) {
      full[i,] <- full[i,]-full[j,]*full[i,j]/full[j,j]  
    }
  }
  full[r,] <- full[r,] / full[r,r]
  
  #setting up a vector with the response, the last value, the one already reduced.
  
  res <- numeric(r)
  res[r] = full[r,c]
 
  # reducing the last part of the matrix 
  for (k in (r-1):1) {
    t = 0
    for (m in (k+1):r) {
      s = full[k,m] * res[m]
      t = t + s
    }
    res[k] = (full[k,c] - t) / full[k,k]
  }
  res <- round(res,2)
  return(res)
}
A = matrix(c(1,1,3,2,-1,5,-1,-2,4), nrow=3, ncol=3, byrow=T)
b = matrix(c(1,2,6),nrow=3, byrow=T)
gauss(A,b)
## [1] -1.55 -0.32  0.95