Data 605 Discussion 3

# Define the function Z: U → V
Z <- function(u) {
  return(0 * u)  # Multiply u by the zero vector in V
}

# Verify additivity
u1 <- c(1, 2, 3)  # Choose a vector u1 from U (example values)
u2 <- c(-2, 1, 4)  # Choose another vector u2 from U (example values)
if (all(Z(u1 + u2) == Z(u1) + Z(u2))) {
  print("Additivity property holds.")
} else {
  print("Additivity property does not hold.")
}

## [1] "Additivity property holds."

# Verify scalar multiplication
u <- c(3, -1, 2)  # Choose a vector u from U (example values)
alpha <- 2  # Choose a scalar alpha (example value)
if (all(Z(alpha * u) == alpha * Z(u))) {
  print("Scalar multiplication property holds.")
} else {
  print("Scalar multiplication property does not hold.")
}

## [1] "Scalar multiplication property holds."

Conclusion, Z is a “stupid” transformation.