Problem Set One

  1. Calculate the dot product u.v where u = [0.5; 0.5] and v = [3; -4]
U = c(.5,.5)
V = c(3,-4)
crossprod(V,U)
##      [,1]
## [1,] -0.5

2.What are the lengths of u and v?

vectorlength = function(x){
  return(sqrt((x[1]**2)+x[2]**2))
}
vectorlength(U)
## [1] 0.7071068
vectorlength(V)
## [1] 5

3.What is the linear combination of 3u - 2v?

(3*U) - (2*V)
## [1] -4.5  9.5

4.What is the angle between u and v?

vectorangle = function(U, V){
  return( acos( (crossprod(U,V)  /  ((vectorlength(U)) * (vectorlength(V) ))))[1] * (180/3.1416))
}
vectorangle(U,V)
## [1] 98.12987

Problem Set 2

Problem set two; disregarding degenerative equations and zero pivots, create a function to solve a 3 variable, 3 constraint system.

A = matrix(c(1,2,-1,1,-1,-2,3,5,4),nrow=3,ncol = 3)
b = c(1,2,6)
solve = function(A){
  # Combine into one matrix
  A = cbind(A,b)
  
#Perform Gaussian Elimination
  multiplier = (A[2,1]/A[1,1]*(A[1,]))
  A[2,] = A[2,]-multiplier
  
  multiplier = (A[3,1]/A[1,1]*(A[1,]))
  A[3,] = A[3,]-multiplier
  
  multiplier = (A[3,2]/A[2,2]*(A[2,]))
  A[3,] = A[3,]-multiplier
  

  l = (A[3,4]/A[3,3])
  h = (A[2,4]-A[2,3]*l)/A[2,2]
  m = (A[1,4]-A[1,3]*l - A[1,2]*h)/A[1,1]
  
  return(c(m,h,l))
  
}
solve(A)
##          b          b          b 
## -1.5454545 -0.3181818  0.9545455