projectData <- function(X, U, K) {
  
  #   Computes the reduced data representation when projecting only
  #   on to the top k eigenvectors
  #
  #   Z <- projectData(X, U, K) computes the projection of
  #   the normalized inputs X into the reduced dimensional space spanned by
  #   the first K columns of U. It returns the projected examples in Z.
  #
  
  # You need to return the following variables correctly.
  Z <- matrix(0,dim(X)[1],K)
  
  # ========================= YOUR CODE HERE =======================
  # Instructions: Compute the projection of the data using only the top K
  #               eigenvectors in U (first K columns).
  #               For the i-th example X[i, ], the projection on to the k-th
  #               eigenvector is given as follows:
  #                    x <- t(X[i,])
  #                    projection_k <- t(x) %*% U[, k]
  #
  
  
  
  # ================================================================
  return(Z)
}