Problem Set 1

\[u = \left[ \begin{array}{cc}0.5 \\0.5 \end{array} \right]; \ v = \left[ \begin{array}{cc}3 \\-4 \end{array} \right]\]

Dot Product

Calculate the dot product \(u \cdot v\)

(dot_prod <- sum(u * v))
[1] -0.5

Length

What are the lengths of \(u\) and \(v\)?

(len_u <- sqrt(sum(u * u)))
[1] 0.7071068
(len_v <- sqrt(sum(v * v)))
[1] 5

Linear Combination

What is the linear combination \(3u - 2v\)?

3 * u - 2 * v
[1] -4.5  9.5

Angle

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

\[\cos \theta = \frac{u \cdot v}{||u|| \ ||v||}\]

(theta <- acos(dot_prod / (len_u * len_v)))
[1] 1.712693

Problem Set 2

elim_solve <- function(A, b) {
  #find pivot
  pivot_1 <- A[1, 1]
  
  #swap if pivot == 0
  if (pivot_1 == 0) {
    A <- A[c(2, 1, 3), ]
    pivot_1 <- A[1, 1]
    if (pivot_1 == 0) {
      A <- A[c(3, 2, 1), ]
      pivot_1 <- A[1, 1]
      }
  }
  #multiplier & eliminate row 2
  mult_2 <- A[2, 1] / pivot_1
  A[2, ] <- A[2, ] - mult_2 * A[1, ]
  b[2] <- b[2] - mult_2 * b[1]
  #multiplier & eliminate row 3
  mult_3 <- A[3, 1] / pivot_1
  A[3, ] <- A[3, ] - mult_3 * A[1, ]
  b[3] <- b[3] - mult_3 * b[1]
  
  #find pivot
  pivot_2 <- A[2, 2]
  
  #swap if pivot == 0
  if (pivot_2 == 0) {
    A <- a[c(1, 3, 2), ]
    pivot_2 <- A[2, 2]
  }
  
  #multiplier & eliminate row 3
  mult_3 <- A[3, 2] / pivot_2
  A[3, ] <- A[3, ] - mult_3 * A[2, ]
  b[3] <- b[3] - mult_3 * b[2]
  
  #solve for x3
  x3 <- b[3] / A[3, 3]
  
  #substitution
  x2 <- (b[2] - A[2, 3] * x3) / A[2, 2]
  x1 <- (b[1] - A[1, 3] * x3 - A[1, 2] * x2) / A[1, 1]
  
  #results
  x <- matrix(c(x1, x2, x3), nrow = 3)
  x
}

Test the system \[\left[ \begin{array}{cc}1 & 1 & 3 \\2 & -1 & 5 \\-1 & -2 & 4 \end{array} \right] \left[ \begin{array}{cc}x_1 \\x_2 \\x_3 \end{array} \right] = \left[ \begin{array}{cc}1 \\2 \\6 \end{array} \right]\]

eq <- matrix(c(1, 1, 3, 2, -1, 5, -1, -2, 4), nrow = 3, ncol = 3, byrow = TRUE)
constr <- matrix(c(1, 2, 6), nrow = 3)

round(elim_solve(eq, constr), 2)
      [,1]
[1,] -1.55
[2,] -0.32
[3,]  0.95